April 2010

This week we have a different type of problem. In order for students to solve a problem in a given lesson, they often need an understanding of and fluency with prior concepts, procedures, definitions, or theorems.  If the needed concepts or procedures are missing, the path can be closed to them. Equally important, in order to support their students’ discovery of a successful path, teachers, parents, and tutors also need to know what prior knowledge and skills are required to solve a particular problem.

So for this week, what concepts and procedures does a student need in order to prove that the distance around the inner object is shorter than the distance around the outer object? We will post some suggestions next week. In the meantime, post some yourself by leaving a comment.

1. Put your left index finger on zero

2. With your right index finger, add 8 (Do not touch zero. Touch 1, 2, 3, 4, 5, 6, 7, 8.)

3. Say, “I just added 8.”

4. Slide your left finger from the zero to the 8 and say, “That’s one group of 8.”

5. With your right index finger, add 8 more (Touch 9, 10, 11, 12, 13, 14, 15, 16.)

6. Say, “I added 8 two times.”

7. Slide your left finger from the 8 to the 16 and say, “That’s two groups of 8.”

8. With your right index finger, add 8 more (Touch 17, 18, 19, 20, 21, 22, 23, 24.)

9. Say, “I added 8 three times.”

10. Slide your left finger from the 16 to the 24 and say, “That’s three groups of 8.”

11. How many times did you add 8? How many groups of 8 is that? How many is in each group? At what number did you end? So 3 x 8 = ?”

How many are in each group? How many groups are there?  How many are there altogether?

12. Try an experiment: Start at zero and add three, eight times. What do you notice?

Second-grade classes are usually taught to skip-count by twos, fives, and tens as preparation for learning to multiply and divide. That is, the teachers give instructions with that goal in mind, but typically not all of the students reach the goal. Some of them memorize the words, but fail to understand the idea behind them, which leads to failure with multiplication and division when they are presented. And some do not even memorize the words correctly. Why is that? I have observed several situations which shed light on this question, which I would like to share with you.

Consider the experience of Jose, a seventh-grader who wanted to show his study hall teacher how fast he could count by twos from zero to twenty-four. He enthusiastically began: “Two, four, six, eight, ten…, eleven?, no…, twelve?, no…, thirteen?, no…” He asked if he could begin again. He seemed to think that—like riding a bike up a steep hill—if he could get going really fast, the right words would flow and he would make it to the top (to twenty-four). He started over several times, but never got past ten, where his memory always failed him. He apparently had no idea how those counting words came to be in the first place; so with no concept of what he was doing, he could not extend the pattern. He could, however, quite successfully repeat from memory a limited number of words that had no meaning to him.

Consider the case of a high-school special ed student, who was attempting to count objects that were grouped in fives. She began, “Five, ten…,” and then went blank. The teacher’s aide mouthed the first sound of the next word in the pattern, to cue her next response: “Ffff…” “Ff…four,” said the girl. “No. Five, ten, ffff…,” intoned the adult. “Ff…fourteen?” replied the child. She could not successfully repeat from memory a limited number of words that had no meaning to her.

And there is the case of Robin, a sixth-grader who recognized that an array of objects was arranged in groups of five, and counted them like this: “Five, ten, fifteen, twenty, twenty-two, twenty-four, twenty-six, twenty-eight!” She understood that she should be skip-counting by fives, but did not realize that she had suddenly switched from counting by fives to counting by twos. A similar thing happened with a middle-school boy, who counted, “Five, ten, fifteen, twenty, thirty, forty, etc.” He unknowingly switched from fives to tens.

I had two experiences myself with memorizing as a high-school student. Our cheerleaders came back from a special summer camp all fired up to teach us some new cheers. The first one they taught us was composed entirely of nonsense syllables. It was crazy, funny, and we could say it! What did it mean? Nothing. Not less crazy was the day that my Algebra teacher demanded that our class memorize a formula, which she wrote on the board: “y = mx + b.” I raised my hand and asked what it meant. “Don’t worry about that,” she said. “Just memorize it now, and we’ll learn what it means later.” Like the nonsense cheer that we learned during the pep rally, we could say the formula. And what did it mean? Nothing. It was crazy, but it wasn’t fun.

Memorizing musical words can be quite charming and engaging—although equally nonsensical. Many Beatles fans can still sing from memory the French words to “Michelle,” but they don’t know what they mean. Thousands of elementary school children enthusiastically sing Frère Jacques every year without understanding the lyrics they repeat. They have fluent recall of the words, but cannot use them to develop their understanding of the language. In fact, millions of American children annually memorize the words to the Pledge to the Flag—which is in English—but they don’t know what many of those words mean, either. And when they proudly recite, “…and to the republic for Richard stands,” they don’t think to ask who Richard is.

I once witnessed an entire class of kindergartners perform a feat of instantaneous linguistic mimicry. Their music teacher told them to listen quietly while he sang them a new song. To his surprise, they started singing the new song right along with him—even though they had never heard it before. As the words of the song rolled out of his mouth, they also came out of the children’s mouths! Somehow those little five-year-olds were able to instantaneously imitate both the words and the tune of the new song! Did they also understand the meaning of the words? Not necessarily. Just as reading comprehension does not automatically follow accurate decoding (reading a sentence aloud correctly, without understanding its meaning), the universal talent for linguistic mimicry and memorization does not always connect with the mental realm of curiosity, meaning, and understanding.

What do the stories of Jose, Robin, and the other students have in common? They were all skip-chanting, not skip-counting. Their words were mere regurgitations of memorized linguistic patterns—like Frère Jacques—not indicators of perceived amounts. They recalled them the same way that students recall the words to the Pledge to the Flag, through sheer rote memorization—just as I had memorized the nonsense syllables at my high school pep rally. But without an understanding of the concepts behind them, fluently recalled words do not provide a sure basis for developing mathematical understanding. Unfortunately, classrooms throughout the country regularly engage in whole-class unison skip-chanting. When all the students are saying the right words together, the teachers are too often satisfied with the deceptive evidence that learning has taken place—not realizing that if only one student in the class can count by fours, the rest of the class can simultaneously imitate the words, just as the kindergartners instantaneously mimicked the words of their new song.

However, rote memorization is only one of many kinds of memory. Just as there are many different learning styles or modalities, there are also many ways of developing memory. I have found that the model most applicable to the development of conceptually connected mathematical (not merely linguistic) memory is the way in which we learn our way around a new city. We do not use flash cards, timed tests, rhymes, tricks, and unison chanting to accomplish that task. Instead, we look at maps and travel repeatedly around the city, gradually building a mental map of landmarks, streets, and locations. And as we continue traveling to meaningful destinations, our need for maps and directions gradually diminishes until we know our way around with such familiarity that we can go where we want without even thinking about it.

This everyday approach to memory development is normally referred to as remembering—as opposed to memorizing. Even though they both have to do with memory, they are clearly not the same thing. Memorization is purely verbal, sometimes has no connection with experience or conceptual understanding, and is much more stressful. Remembering, on the other hand, is generally derived from multi-sensory kinesthetic experiences within a spatial/conceptual context, with a connected verbal component, and generates relatively little stress.

Here is an example of how to teach skip-counting by fives in a way which incorporates a mathematical way of remembering, rather than a linguistic way of memorizing. Present the students with their own copy of a vertical array of pea pods, with each pod containing five numbered peas.

[   (1)     (2)     (3)     (4)     (5)  ]

[   (6)     (7)     (8)     (9)    (10) ]

[  (11)   (12)   (13)   (14)   (15) ]


Giving very brief directives, accompanied by clarifying gestures…

• Have the pupils touch each pea and count them aloud in a quiet voice—except for the last pea on the right in each group, which should be counted in a loud voice:

“One, two, three, four, FIVE;

six, seven, eight, nine, TEN; etc.”

Watch the students carefully, and do not allow the class to go faster than the slowest student can manage.

If the speed accelerates to a pace comfortable only for the fastest students, the slower ones are likely to stop touching-and-counting, and will cope with the situation by reverting to rote verbal imitation of their peers.

When the special ed student mentioned above was given these directions, she followed them easily, then smiled and said “I get it!” It is important that the objects (the peas, in this case) in such an array be numbered; without that visual imprint, some students do not make the connection between the numbers that are saying aloud and the written numbers that teachers hope they are imagining (because they are not imaging them!).

• Next, have the students whisper the counting words as they touch the peas, saying each fifth number in a regular voice:

“One, two, three, four, five;

six, seven, eight, nine, ten; etc.”

When the girl mentioned above followed these directions, she laughed and said “This is easy! I can do it!”

• Then have the pupils touch-and-count the peas again, thinking the first four counting words as they touch the peas, and saying each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The girl’s reaction conveyed the pleasure engendered by understanding: “This is fun! I like learning this way!”

• Then have the pupils slide a finger across each group of peas, smoothly gliding from left to right as they again think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

• Now present the students with another vertical array of pea pods, with each pod containing five unnumbered peas (see graphic above).

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]


Then have the pupils again slide a finger across each group of peas, as they think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The students are now skip-counting by fives. The same approach can be used to teach skip-counting by other numbers—only different graphics are required. Each child must be given the opportunity to show that they can do the last step (skip-counting groups of unnumberd objects) by themselves, without the possibility of verbally imitating other students. The first three steps should be reviewed (with teacher support) by students who display uncertainty, hesitation, or inaccuracy. (“You sound like you’re not certain about that. Count them for me again so we can see if you’re right.”)

What distinguishes skip-counting from skip-chanting? Skip-counting requires physical interaction with numbered objects; skip-chanting requires no actual counting at all. Skip-counting requires mathematical thinking; skip-chanting can be accomplished with no thinking whatsoever. Many teachers rely on slow-paced counting activities with physically constructed grouped amounts to develop the concept of skip-counting, but then revert to skip-chanting to develop fluent recall of the number-facts. They assume that since their students understood the conceptual part of the lesson, they will automatically invest the following rote verbal exercise with understanding. Experience shows that some students do make that connection. Other students, however, are not rapidly recalling the previously counted groups during the group chanting exercise; for them, skip-chanting is a near trance-like state with only one goal: successful verbal duplication. Verbal imitation does not require any conceptual connection in order to be successfully performed. For that reason, some students make the connection while others do not; during the same group activity, some students can be truly skip-counting, while others are merely skip-chanting. The pea-pod lesson modeled above provides all students with the means to remain conceptually connected during the memory-building phase of the skip-counting lesson.

Mathematics presents students with a mode of thinking/reasoning. It includes observation, attention to detail, analysis, synthesis, relevant question asking, and problem solving.  It involves some valuable traits like the ability to handle sweat, frustration, dead ends, perseverance, and the discovery that there is wonder, joy, and even some exhilaration at the end.  We invite students deeper into or higher up this mode of reasoning year-by-year, subject-by-subject. So what is higher level mathematical reasoning?

A look at some of the approved and adopted texts suggests that a typical answer is, “algebra.” Algebra is generally considered to be higher level math thinking for today’s school students. Constants, variables, coefficients, expressions, equations, quadratic equations, real, rational, and irrational numbers, and combining like terms…  If we can just get upper and even lower elementary students to start thinking about some of this, we believe that there is more of an opportunity for higher level math reasoning.

But what about geometry students who have already passed Algebra I, but still have not mastered basic number sense concepts involving fractions? For example, I tutored a high school geometry student recently who did not realize that if amount A is half as much as amount B, then amount B must be twice as much as amount A. This student had memorized the formula for determining the measure of an inscribed angle (it is 1/2 the measure of its intercepted arc), and had solved many problems correctly. But when asked to find the measure of the arc when given the measure of the angle, the student was stumped. It seems that for this student, thinking about basic fractional relationships was actually higher level mathematical reasoning—higher than the current level of understanding.

Higher level math reasoning for students is simply whatever the next step is from where they are now. The relationship between 1/2 and twice, or that a group can be both one and many, or that a “1” sitting in the tens column has a different value than a “1” in the ones column are all higher level math thinking for students who do not yet understand those concepts. People generally consider algebra more abstract than arithmetic, because it appears to be less concrete—and therefore it must be the flagship of “higher level mathematical reasoning.” But any concept is “abstract” to the student who does not understand it yet!

The critical element is not the level of difficulty of the work, but whether or not the work is being addressed through reasoning. Students who can factor quadratic equations because they have memorized a bunch of rules cannot be said to be applying higher level mathematical reasoning, unless they actually understand why they are doing what they are doing. There is a big difference between “higher level activities” and “higher level mathematical reasoning.” When higher level activities are taught through mere memorization or repetitive activities devoid of real understanding, they do not involve any reasoning at all. When lower level activities are taught in ways that make students really think, then those students are involved with higher level mathematical reasoning. And math teaching need not hang its head and feel inferior to other academic disciplines while focusing on these lower level activities.

Algebra is not the problem in itself. Thinking that it accomplishes the need for higher level reasoning and application is.

Another unfortunate answer to what is higher level mathematical reasoning can be seen in the rush to complicate problem sets in textbooks. The geometry book that the student I tutor is using in school, published by a major publisher and state adopted, has outstanding higher level math reasoning problems to solve. I’m having as much fun with some of them as I’m sure that authors and state committee members had. But my student and many in her class are not. There are precious few problems in any section of this book that allow students to develop a confident understanding of the basic concepts and procedures before “higher level math reasoning” is introduced in the form of clever and complicated levels of application.

Rather than leaping to higher level activities that require fluent reasoning that has not yet been developed, the interests of students would be better served if this book (and others like it) presented step-by-step contexts of problems of graduated difficulty—each problem based on the reasoning developed in the previous problem, and preparing students for the next step of reasoning represented in the following problem. The proper function of a math book is to develop mathematical reasoning, not merely to create problems that require its use. By rushing to over-complicate the problems, textbooks unwittingly exclude many students from success, actually thwarting the development of their reasoning and forcing them to rely on mere memorization to cope with their work.

Yes we need to keep earlier concepts and procedures alive by integrating them into problems in subsequent chapters, and yes students need to explore multiple uses and applications, and yes they need to use all of this to solve mathematical problems and not merely perform arithmetic calculations. I am not arguing against any of this. But enrichment is enriching and higher level mathematical reasoning is only reasoning when students have access to it. We should take as much pride in opening up and developing that next level of higher mathematical reasoning, whatever it may be, as we do in the creative, clever, complicated, and fun problems our mathematical minds conceive. We should remember what it’s like for those who are new to all of this. What is higher level mathematical reasoning for them?

When learning to multiply a single-digit number by another single-digit number, students sometimes don’t understand the concept, and sometimes have trouble remembering the facts. Problem of the Week #11 provides a “manipulative on paper” that can be used by these students to establish an unknown multiplication fact in a conceptual way.

Here is the challenge:  Using only the Number Line shown below and the index finger from each hand (no writing or coloring!), find the product of 3 x 8 in a way that incorporates the “Three How Many’s of Multiplication:” 1. How many are in each group?, 2. How many groups are there?, 3. How many are there altogether?

The solution will be posted next week!

When learning to divide a three-digit number by a one-digit number, students are often only given a mechanical procedure [or abstract algorithm] to follow, without a concrete picture or objects to manipulate to establish a physical sense of comparison and magnitude.

Problem of the Week #10 gives the opportunity to picture a three-digit number divided by a one-digit number problem using a special array of “manipulatives on paper.” This is actually quite easy to do when you know how; but like last week’s multiplication problem, most people have never done it before.

Here is the challenge: imagine that you have never been shown how to do long division. Using only the “One-Dollar Bills Page,” the “Ten-Dollar Bills Page” we used for last week’s problem, and two blank pieces paper, show an exact picture of 156 ÷  4. Your solution must not only show the correct amount of money, but it must picture the actual meaning of 156 ÷  4. The solution will be posted next week!

Hint #1. Using the “One-Dollar Bills Page” and two pieces of blank paper, show 4 people each having $3.

Hint #2. Using the “Ten-Dollar Bills Page” and two pieces of blank paper, show 4 people each having $50.

#3. Put the “One-Dollar Bills Page” on top of the “Ten-Dollar Bills Page.”  Move the “One-Dollar Bills Page” to the right, so that each person on the “Ten-Dollar Bills Page” has $50.

#4. Put a blank piece of paper on top of the “One-Dollar Bills Page,” so that each row on the “One-Dollar Bills Page” has $3.

#5. Put a blank piece of paper cross-wise on top of both money pages, so that only the top 4 rows of $53 are showing. This is a picture of 53 x 4 (4 people, each having $53).

Here is our podcast on the progress chart as a way to both measure and purposefully improve fluency and understanding. See our post, A Model for Using Guided Discovery in a Math Lesson, for when to use it in an effective lesson.

Vodpod videos no longer available.

more about “The Progress Chart“, posted with vodpod

As we wrote in previous posts, fluency is important. Basic math facts do not go away but are used throughout the upper grades in more abstract and involved contexts. If the students’ minds are burdened with having to reestablish basic facts and concepts, they’ll be like music students not fluent with their basic scales, burdened and frustrated later when they come across those scales in a piece of music.

The progress chart, found in the middle of every partner page, builds on the good practice started with the graphic and partner pages. It uses those pages to keep practice grounded in experience, so there are no tricks or drills separated from a physical context that is understood. It takes what is good about timing exercises and leaves behind the problems. It both measures and improves performance. It develops further the understanding, speed, and accuracy needed for work in the upper grades.

The overall time and increments in the progress chart differ between lessons based on our experience in the classroom and tutoring. You may find it helpful to read the earlier posts in this series, A Model for Using Guided Discovery in a Math Lesson, if you haven’t already, as these steps are sequential.

Here’s a model for working with the progress chart.

1. After students are oriented and comfortable with a graphic and accompanying partner page, ask for a volunteer to do a timing experiment with you. While they say the answers on the partner page, time them to see how many seconds it takes them to get a perfect score (with the help of the graphic page if needed). You can say that it’s their first time doing this so it doesn’t matter if they’re fast or slow; this helps relieve a sense of anxiety. To avoid the tension sometimes associated with racing, do not begin timing with “ready, set, go!”; simply begin the timing when they say the first answer. Their partner should continue checking to make sure the answers are correct as explained in the post on the partner pages. When they finish, tell them their time and have them circle all the times on the progress chart that they beat.

There is no target time. However long it takes to get a perfect score is the objective at this point. Because of their previous work with the graphic and partner pages, many students will move comfortably into this timed approach. The problems are better understood and they’re becoming more familiar to them. To keep this from being a rote exercise, change the direction in which they solve the problems.

2. Now ask the class to look at their partners and decide who is partner A and who is partner B. When you say “begin,” all the A’s will say the answers and the B’s will check the answers and keep track of the time with you. When they’re done, they can look up and you’ll give them their time.

3. The times that are beat are circled and the partners switch roles.

4. The partners continue taking turns, comfortably and joyfully improving their times by increasing their familiarity with the material. After the initial work on the graphic page, reinforced and expanded on with the partner page, the progress chart creates even greater competence and confidence with the concepts, procedures, and facts.

This is very different from enforced time tests, where students do as many problems as they can in a predetermined time—which often means that they fail to answer some problems, and answer others incorrectly. Our emphasis is on correctness, thoroughness, and understanding every bit as much as time. The students correct themselves as they go; all of the problems are answered correctly. Therefore, no wrong answers are written and reinforced, and no answers need to be given to them. They don’t miss needed practice on certain problems because of not finishing the exercise in a predetermined time. The voluntary character of working with the progress chart creates a happy feeling. And our experience continues to be that students’ natural inclination is to keep going to improve their time, and in so doing, their understanding and fluency.

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