Traditional Math (6): Negative Numbers for Seventh Grade (cont) — Multiplying and Dividing Negative Integers

This is the conclusion of the chapter on negative numbers for seventh grade. These chapters will eventually become a book on Traditional Math, to be published by John Catt Educational.

  1. Multiplication of Negative Integers

The multiplication of negative integers can be a confusing topic for students—particularly the rule that the product of two negative numbers equals a positive number. The main problem that students have is in seeing what multiplication by a negative number may mean.

I’ve found that providing examples of situations modeled by multiplication with negative numbers is effective in helping students understand the rules. I start with a review of negative and positive numbers in terms of changes in particular situations. That is, their daily experience with integers is how they describe things like temperature, electrical voltage, elevation above or below sea level, bank balances, and gains and losses.

Integers also can represent a change in the situation as they have seen with the “gains and losses” problems. That is, changes in money earned versus money lost, temperature increases and decreases.

To start, I ask students to describe various changes as positive or negative”

  1. a) Ann gained 4 pounds in the last month (4)
  2. b) Jerome lost 14 pounds in a week. (14)
  3. c) Kathy lost $40 on a roller coaster ride. (-40)
  4. d) Five minutes from now (5)
  5. e) Ten minutes earlier (-10)

The next examples require a bit more thought; describe the change in terms of positive or negative:

  1. f) The temperature changed from -3 degrees to 2 degrees.
  2. g) The football team lost 5 yards on the first play and gained 10 yards on the second
  3. h) A bird was at 120 feet above the water to 30 feet above the water
  4. i) The water was turned on at 10:00 AM and turned off at 12 noon.
  5. j) Janet finished the drive at 3:00 PM; she started at 1:00 PM.

Multiplication of two numbers in which one number is negative. Students know how to multiply positive numbers and know how to represent them as repeated addition. A problem like “Ted made $10 an hour for 3 hours; how much was his total pay?” is represented at 10 + 10 + 10 or 3 x 10. 

After showing the above problem I ask students how they would write the following problem using repeated addition: “Sonia lost 3 pounds for 2 weeks in a row; how much did she lose after two weeks?” (If students need a hint, I will ask how they would represent a loss of 3 pounds.) Students will generally know the answer intuitively and upon hearing the answer of -6, I write on the board:

(-3) + (-3) = 2 x (-3) = -6.

 Finding an example problem to represent (-2) x (3) is a bit more difficult since adding three -2 times does not make sense. Dolciani’s “Modern Algebra: Structure and Method” (1962) contains an example that I’ve used in seventh grade which provides meaning to the negative values.

The example is of a water tank with water flowing into it at the rate of three gallons per minute. I ask if the three gallons per minute is a positive or negative number. If it flows at three gallons for every minute, how much more water will there be in the tank after two minutes? Notice I am asking how “much more”, not what is the total amount of water. We are calculating the change in the amount of water in the tank and students are quick to give the correct answer of six. I now ask if the time, two minutes, is positive or negative.  Both numbers are positive and we can represent the situation as 2 x 3 or 6 gallons more water than what was there before. This suggests the rule they all know:

2 x 3 = 6:  Positive number x positive number gives positive number.

For the next scenario, I want to know how many gallons less was in the tank two minutes ago, if water is flowing into the tank at three gallons per minute. Intuitively, most students will know that there will be six gallons less, because the water in the tank is increasing for each minute. Therefore for each minute prior, there was three gallons less. In my experience students will shout out this answer. I ask how “2 minutes ago” is represented. I will hear someone, usually hesitant, saying “Negative two?” And that is correct. We know there will be six gallons less than there was, so we have:

(-2) x (3) = -6: Negative number x positive number gives negative number.

This example shows that the negative number can be the multiplicand (i.e., the number being multiplied) and the positive number the multiplier.

Now I tell the students to assume that the water is flowing out of the tank at the rate of three gallons per minute. I ask how we represent that, making sure they understand that it is -3, since it is representing a loss. The problem now becomes what is the change in gallons after two minutes. Students should recognize that the two minutes is a positive value.  I ask how we represent this situation as a multiplication statement. Since this is similar to the opening problem about losing 3 pounds per week for two weeks, students should that the answer is once again 2 x (–3) = -6. This time the -6 represents a loss of 6 gallons. This problems suggests

2 x (-3) = -6: Positive number x negative number gives negative number

The final example is a negative number times a negative number. Before I present the example, I ask if anyone knows whether the product will be negative or positive.  For those who say it’s positive I will ask why. Some explanations will be vague, but I’ve found that at least one person will extrapolate what we’ve done before. In such instance I then proceed with the example. If it doesn’t happen, it isn’t a problem. I just proceed and say something along the lines of “Let’s find out.”

In this case, the water is flowing out of the tank at the rate of three gallons per minute, represented as -3.  I want to know if two minutes ago (-2) there was more or less water in the tank, and by how much. At this point, most will know the tank held six gallons more water. I now write:

(-2) x (-3) =6: negative number x negative number gives positive number.

 At this point the rules are summarized:

  1. The product of two integers with different signs is negative.
  2. The product of two integers with the same sign is positive.

Admonition. After establishing the rules via the examples, I admonish students that the examples only suggest that these rules are true. There is a mathematical proof of these rules which I provide after they have learned about the distributive rule. For now, it’s all they need in order to get a sense of what’s happening when we perform these multiplications.

Nevertheless, there will undoubtedly be some students who will not understand how the examples work and why the rules are suggested by them. I tell these students that it will become clearer the more they work with such problems, but that for now should just follow the rules and “trust the math” that it is telling a true story.

Practice and extension. Students are now ready to work on guided practice problems, the first few of which summarize what we have just learned—problems such as (-5) x (-7), (-2) x (4), (-1)(5), and (-1)(5).  After the last two problems I will give them the following:  5 + (-1)(5), and -5 + (-1)(-5). The first problem becomes 5 + (-5), which goes back to what they have just learned about adding negative integers. It is not unusual that they will look at the problem as if they have never seen it before.  I ask them if they can remove the parentheses, and remind them that +(-) equals -.  The problem then becomes 5 -5, or zero.

Similarly -5 + (-1)(-5) becomes -5 + 5 which is also zero.  I will ask them to summarize what multiplying a number by -1 changes it into. There may be blank stares, and if so, I will remind them that 5 and -5 are called “opposites” which some have likely forgotten. It is not unusual for such lapses in memory, given the amount of new information that they are taking in.  But it is important to link what they have learned in this lesson with what they know about addition and subtraction rules. To that end, other problems may include: 5 – (-2)(5); -4 + (3)(-2), and so forth.

So for 5 – (-2)(5), the number (-2) is multiplied by 5 which yields -10. The resulting problem is then 5 – (-10). From this point, a common mistake will be forgetting to include the negative sign next to 10, and writing 5 -10. 

For these type of combination multiplication and addition/subtraction problems, it will be necessary to remind them of order of operations which they have had in sixth grade—multiplication operations are performed before addition or subtraction. Also, using parentheses to denote multiplication should be explained. This notation is easier to work with than the form 5 + (-1) x (5), since the parentheses make the order of operations more obvious. In addition, using parentheses for multiplication prepares students for algebraic notation. Later when they work with evaluating expressions like a + bc, by substituting numbers for a, b, and c, they will already have had experience with that form. Also, they will see that inan equation in the form 2x +4 =8, the 2x represents 2 multiplied by an unknown number.

Multiplying more than two numbers.  Before I set them loose on homework, I put up a problem and ask how I would solve it:  (-2)(3)(-5). Those who get it, I then ask to explain to the class. Such problems are broken down and solved by multiplying two numbers at a time; they above problem then becomes (-6)(-5), which is 30.  A problem like (-5)(2)(-2)(4)(-1) becomes (-10)(-8)(-1), which is -80.  After maybe two more, I will ask them if the number of negative numbers in the problem helps them determine whether the final product is negative or positive. I have them discuss that, leading them to see that an odd number of negatives will result in a negative product.

  1. Division of Negative Integers

It is definitely recommended that the warm-ups preceding today’s lesson include not only problems about multiplying negative integers, but also addition and subtraction. New information recently learned tends to eclipse older information. This can be seen by a common mistake that students will start making, and I’m sad to say will persist among some students well into the school year. That mistake is to conflate the rule stating that the product of two integers with the same sign is positive with the rule for addition of two integers with the same sign. Students will see a problem like -5 -5 and now in addition to making the mistake of saying it is zero, will then say it is positive 10. 

Warm-ups should include straight multiplication problems like (-2)(5) and (-1)(-4)(3), but also a problem like 2 – (-2), 2 + (-5), and the combination multiplication and addition/subtraction problems that get at the problem described above.  For example -5 – (-1)(-5), which becomes -5 – 5. 

Going over the warm-ups will then serve as a review of what has been covered so far. At this stage I have found myself saying “Why did I ever think this was going to be easy?” Some classes will get it more easily than others, I’ve found, but there are always relapses, forgetfulness, and it will be necessary to repeat the rules.

Division as inverse of multiplication. Division of negative numbers is a straightforward application of the rules for multiplying negative numbers. The rules are the same:

  1. The quotient of two integers with like signs is positive.
  2. The quotient of two integers with unlike signs is negative.

I start the lesson by asking what 2 x (-3) is.  Hearing -6, I then write 2 x (-3) = -6.  Since division is the inverse of multiplication we can divide the produce, -6 by either 2 or -3 to get the other factor. That is -6 ÷ (-3) = 2, and -6 ÷ 2 = -3.  In other words, when we divide -6 by 2 we are seeking a number which when multiplied by 2 yields -6.  That number has to be negative. I will ask the class why and then wait while an uncomfortable silence pervades. I would like someone to say “To get -6, 2 has to be multiplied by -3 because a positive number multiplied by a negative number is negative.” I will settle for something reasonably close, however. If someone says that -3 x 2 equals -6 I’ll go with it and give further examples. I don’t want to spend a lot of time trying to get them to say the right thing and then have them forget what it is we’re learning.

Similarly if we divide 6 by -3, the number that is multiplied by-3 to yield 6 has to be negative. Why? Because the product of two negative numbers is positive.

I will then provide examples for the class to work on and include combination problems such as -10÷2 -3. Again, they will need to be reminded of order of operations; i.e., division operations occur before addition or subtraction.

The bottom line rules are exactly the same as multiplication: for two numbers which can be paraphrased into something succinct enough to fit on a bumper sticker: Like signs: positive. Unlike signs: negative.

Despite the fact that the rules are ultimately the same, students will still get confused. It is a matter of repetition and practice. Also, it is essential to include all types of problems that involve computations with negative numbers, not only in warm-up questions but in quizzes and tests given throughout the year.

I conclude this sub-chapter by providing examples of warm-up questions that I have given after covering all the topics discussed in this entire chapter.

  1. -3 x 3
  2. (-5 + (-20) ) ÷ (-2)
  3. -60 ÷ (-10)
  4. (5-10) x (4-8)
  5. (5-(-4) ) ÷ (-3) = ? + 6

Problem 5 is a more challenging problem and it is admittedly a “front-loading” of the type of problem they will be solving later when we cover expressions and equations. Typically this problem opens up discussion when we go over the problems despite the hints I gave students when they are working the problems. 

Solving the left hand side results in 9 ÷ (-3) which equals -3.  We now have to find what value the question mark represents.

I will sometimes give them a simpler problem that they’ve seen before such as 9 = 7 + ?.  They will solve this quickly (usually—there are always exceptions!) and hearing the correct answer, I ask how they did it.  The usual answer is “Two added to seven is nine.”  Which is correct, though I’d much rather hear, “Nine minus seven equals two.”  If I don’t hear that, I will say it; something like, “So if 9 = 7 + 2, we can also say 9 – 7 = 2 and 9 – 2 = 7.”  This same pattern can then be applied to the problem at hand : -3 -6 = -9. 

 

 

Traditional Math (5): Negative Numbers for Seventh Grade (cont) — Subtracting Integers

This is a continuation of the chapter on negative numbers for seventh grade. The next installment will be on multiplying and dividing negative numbers.

Sub-Chapter 4: Subtracting Integers

This lesson provides a clarification and, for some, a revelation, that subtraction of two numbers is the addition of an additive inverse. In formal mathematical terms, x – y is defined as x+(-y) where -y is the additive inverse of y. They have seen this already with problems such as 4+(-10) on the number line, which they have learned is the same as 4-10. There is one important case that hasn’t yet been explored, which is subtracting a negative integer; e.g., 5-(-10). This particular case will be the new procedure that they learn; everything up to that point is a clarification of what has come before.

At least one of the warm-ups I’ve used for this lesson provides a segue to the subtraction of a negative number. Specifically:

The temperature yesterday was 4 below zero. Today it is zero. By how much did the temperature increase?

 

Adding the opposite. I start this lesson by asking the class to find 7 + (-4).  After this is done, I think ask them to tell me what 7 -4 equals.  I ask why we obtained the same answer and if I don’t hear something along the lines of “It’s the same problem” (which sometimes happens despite all my intentions) then I’m not afraid to say “Do you suppose that 7 +(-4) is the same problem as 7 – 4?”

At this point I disclose that whenever they have been adding negative integers, they are subtracting. Subtraction is really the addition of an additive inverse.  Stated more simply, subtracting a number is the same as adding the opposite of this number.  I quickly give an example: The problem 7 – 3 is the same as adding the additive inverse of 3, which is 7 + (-3).  I ask them to write it without brackets as we did in the previous lesson: 7 – 3.

Addition and subtraction are what are called inverse operations. The students have been using this fact for years, having been told that 5-2 is a number which when added to 2 equals 3.  That is 5 = 2 + x.  They have worked with number bonds or number families, so they are familiar with 2+3 = 5, 5-3 = 2, and 5-2 = 3.  When we have a problem like 4 – 10 (warm up problem 3), we are finding what number added to 10 equals 4.  They may have solved it using the “good day, bad day” technique, seeing first that there is a loss and it’s a loss of 6, or -6.  I then write 10 + (-6) and ask what it is equal to. They can see the answer is 4.

I remind them that this is how they were taught to check if their answer to a subtraction problem is correct. This may seem like new information even with reminders that this is what they have been doing the past few days starting with the “gains and losses; good day, bad day” technique. Therefore, more examples are necessary to reinforce the procedures so they are comfortable with doing such problems.  I then give them three or four problems to do and have them check the answers. Thus, to check that 3- 6 equals -3, the student would add 6 + (-3) to obtain 3, which checks.

 

What Subtraction Represents.  A problem like 10 – 4 can represent the loss of 4 things—those things being many different items such as money, weight, length and so on. For example, the question “If the temperature was 10 degrees and it decreased by 4 degrees, what is the resulting temperature?” is answered by subtracting 4 from 10. If, however, the question were “The temperature was 4 degrees and now it is 10 degrees. By how much did it change?”, the answer is still six, but the numbers represent different things.

In the first instance, six represents the new temperature after a decrease of four degrees. In the second it is the amount of increase in temperature from four to ten degrees.

Subtracting a Negative Integer. Now we come to subtraction of a negative integer such as 10 – (-4). The second model is what we use when we present subtraction of a negative integer in order to keep things straightforward for students, I limit the examples to finding the amount of change rather than what a loss of -4 represents.  (An example of a problem that asks what’s left after a loss of -4 would be: A person has $10 in his bank account after $4 has been deducted in error. The bank corrects this error by removing the debit of $4. This is done by subtracting the loss of $4: 10-(-4), which then becomes 10 + 4 or $14.  Even adults may find it confusing that ancelling a debt can be represented by the subtraction of a negative number. It is therefore highly likely that seventh graders will find the concept difficult. Since the goal is for   students to subtract negative numbers, it is far easier to explain the procedure using the “find the difference” model discussed above, rather than by the “find what’s left” model.)

I first point out how they answered the question of “If it were 4 degrees yesterday and 10 degrees today, what is the change in temperature?” I want students to see the form they used: Today’s temperature minus yesterday’s temperature.

Next, I ask: “If it were 4 degrees yesterday but 10 degrees today what is the change in temperature?” Using the form defined above, we obtain 10-4 which represents an increase of 6 degrees.

We are now ready to present the problem of 10 – (-4).  The question becomes: “If it were 10 degrees today, but -4 degrees yesterday, what was the change in temperature.” It can also be stated, “What was the increase in temperature from -4 to 10 degrees? The problem can now be written as 10 – (-4), which I leave up on the board.

Drawing a number line (either vertical or horizontal), I plot -4 and 10. As mentioned earlier, one of the warm-up problems for this lesson asked how much must the temperature increase from -4 degrees to reach zero. Although one would hope that students remember the warm-up problem, I usually have to remind them of it, as well as the answer: the temperature must increase 4 degrees to get to zero degrees from -4. That is, -4 + 4 equals zero.

Transferring this to the number line, it is apparent that zero to 10 degrees is an increase of 10 degrees, so we add 4 to 10—an increase of 14 degrees.

I then explain that rather than using a number line to calculate it, we can use the “add the opposite rule.  The problem 10 – (-4) becomes 10 added to the opposite of -4 or 10 + 4, which is 14.

Another example using depth under water illustrates subtracting a negative number as well. I scaffold the problem by first starting with all positive numbers: A bird was 5 feet in the air and flew up to 10 feet. What was the distance upward that it flew?  Students will easily see that it is “new height minus original height”, or 10 – 5.

The problem is now changed so that a bird dives underwater 5 feet to catch a fish, and then flies upward to a height of 10 feet above the water. What was the upward distance that it flew? I provide prompts, such as “How do we represent 5 feet underwater?” (-5) and “How is the problem written?”  10 – (-5).

I also want to show problems where there is a decrease. For example: “The temperature today is -5 degrees; yesterday it was 6 degrees. What is the change in temperature?”  The problem is written as -5 – 6, which equals -11, or a decrease of 11 degrees.

After these worked examples, students are now given four or five problems that require subtracting a negative. One or two problems will be word problems but the others are strictly numerical.

Vertical vs Horizontal Number Lines. Some teachers I have spoken with recommend using vertical rather than horizontal number lines. They have observed that when the number line is placed vertically, the students more easily grasped the idea of subtracting a negative. It appeared that up = positive, down = negative was a “natural” visualization as opposed to left = negative, right = positive. This is something to keep in mind if students struggle with this and other concepts—teachers can move back and forth between vertical and horizontal number lines as necessary.

Common error (again).  Students will continue making the mistake of seeing problems like -4 -4 as zero. Now there are three ways to adjust their thinking.

1) Gain/loss method: “I lost $4 and then I lost $4 more.”

2) Number line method: -4 -4 on the number line is (-4) + (-4).

3) The problem -4 – 4 is not the same as -4 – (-4). The latter equals zero because when one adds the opposite it becomes -4+4, and adding a number’s opposite will always equal zero.

Giving it a rest. Mastering the operations with negative numbers will be confusing for some students at first. Although these lessons continually refer to, build upon and reinforce the “gain/loss” procedure, some students may become overwhelmed by new information as well as the mathematical way of stating things. In particular, they are now learning to think of subtraction as the addition of an opposite number.

It is advisable to give students time to work with the newly learned procedures to ensure that they are comfortable with them and are achieving automaticity. Many textbooks, however, provide an additional topic such as showing pictorially how addition and subtraction work using circles where each circle represents a positive or negative unit integer. The purpose is to spotlight the conceptual underpinning of adding and subtracting negative integers. An example of how it works is shown in Figure 3.

Figure 3. Pictorial approach for subtracting integers, using circles

In my experience, when students are still trying to master the procedures for adding and subtracting negative integers, additional approaches may confuse more than enlighten. Some students may not be ready to absorb the yet another pictorial approach, particularly when they have adjusted to the pictorial approach using the number line.

The pictorial approach can be given later when, in the estimation of the teacher, students appear proficient and comfortable working with negative integers. I tend to not present it, particularly if I see that students are successful in working with negative numbers. Ultimately it is a judgment call, based on a teacher’s experience with teaching it, as well as their understanding of how it works and how and when to present it.

Traditional Math (4): Negative Numbers for Seventh Grade 

This is the first part of the unit on negative numbers for seventh grade.  It includes a general introduction, gains and losses, and adding negative numbers.  Tomorrow will be subtraction, and the next day multiplication and division.

In my Math 7 classes, the beginning of the school year begins with the unit on integers, which includes operating with negative numbers. A new teacher in a new classroom, with new school supplies coupled with a topic students haven’t had before often has the same allure and excitement as that of a shiny new toy that holds great promise for many exciting and life-changing hours.

Students view me, their new teacher, just as they do their brand new school supplies including the graph paper notebook that I hand out on the first day. New notebooks (particularly graph paper notebooks) hold the promise of being filled with information that will make them smart. After the first few lessons, it has not been unusual for me to hear from some parents that their kids have remarked “This is the first time I’ve really understood math.” 

As things become more complex, the feeling of newness and promise fades. Students go from saying “I finally understand it” to “I hate math” sometimes in the span of less than a week’s time. It is also not uncommon for students to understand and carry out a procedure perfectly the day it is introduced, only to have totally forgotten it the next day, with some students asking “When did we learn this?”

Nevertheless, there is good news. Based on what they have been doing in the lower grades with differences of quantities and computing changes in weight, amounts of money and so forth, students at this point know what losses and gains are and how to compute them. We build on this prior knowledge and intuition so that they are able to express and compute quantities in terms of negative numbers.

The first lessons build upon what they have learned previously. As the topic becomes more complex and students become confused, teachers can and should refer back to some of the introductory techniques as a way to underscore that what is being taught today is building on what they perfectly understood just a few days previous.

It is also important to continue refreshing the procedures for operating with negative numbers throughout the year, to ensure what has been mastered stays that way. Continued repetition and practice helps to lock in the procedure and ensure automaticity.

Some aspects of negative numbers will seem abstract to students to the extent that a procedure may not make logical sense or seem counter-intuitive. The mathematician John Von Neumann once said “In mathematics you don’t understand things. You just get used to them.” Like many things in mathematics, after experience and practice with a procedure and concept, what was once alien becomes familiar. The familiarity eventually allows students to see the concept as reasonable and accept it—and they may even wonder why they ever found it confusing. At that stage, it is not unusual for students to say “I understand it now.”  This is particularly true with the topic of negative numbers.

General Overview

The general arc of progression on this topic is:

1. Introduction to negative numbers: Number line, order, and absolute value

2. Gains and losses

3. Adding negative integers

4. Subtracting integers

5. Multiplying and dividing negative integers

1.  Introduction to negative integers

This unit focuses on the mathematical operations of addition, subtraction, multiplication and division with negative numbers. Some students have may have learned the operations with negative integers in sixth grade, depending on what textbook was used, and/or the inclinations and goals of their sixth grade teacher. For those students, this unit will be a review. Others may have had an introduction to what negative numbers are, but not operations with them.

I start this unit with a review of what all students may know about negative numbers. This review is generally not included in seventh grade textbooks and focuses on concepts with which they are familiar. It is an overview of what negative numbers represent, using examples such as temperature (degrees below zero), or depth (feet underground or under water). I generally take two days to do this review and overview.

On the first day, we cover the general concepts of gains and losses, and their representation. A gain of $10 can be represented as +$10 or $10. A loss is represented as -$10. A descent of  feet can be represented by the number -50. A drop in temperature of 10 degrees: -10

Some students will observe that you cannot possess a negative amount of anything. If no one makes this observation, then I do. It is worth mentioning that negative numbers can be a comparison, or a relative amount. For students just learning about negative numbers, it is a new way of expressing comparisons and changes. For example, students will readily answer the problem “If it was 60 degrees yesterday and 40 degrees today, what is the change in temperature?”

I will give students this problem and upon hearing the correct answer of twenty degrees, I ask if it is twenty degrees more or less. Hearing “less”, I then point out that it is a loss of twenty degrees. Since they have learned that a loss can be expressed as a negative number, I then ask if I can say that the change is -20 degrees.

Negative numbers can also be used to indicate direction, or relative position.  I will ask: “In a football game, if your team lost ten yards, how would you indicate that using a sign?” Of course I want them to say -10 and they usually will. The negative number tells us about the position of the ball relative to where it started.

Number Line. The discussion about negative numbers and direction directly relates to the number line, so I introduce it at this point. Students have seen number lines before, but now we look at it with respect to negative numbers.

 There are two simple principles for number lines that I state:

1. Negative numbers are numbers that are less than, or to the left of zero; positive numbers are to the right of zero.  

2. The bigger the number, the farther it is to the right. The smaller the number, the further it is to the left.

A number left of another number is less than that number: 5 < 7,  – 7 < -5.  A number right of another number is greater than that number.  -2 > -8. 

I might ask “If it is -20 degrees today and it was -30 yesterday, which day was colder?” It is obvious that -30 will be colder, and by plotting the points on a number line on the board, students easily see that -30 is to the left of -20.

I will show two numbers to the right of zero on the number line; say 5and 7. When we have two positive numbers, the number furthest from zero is the greatest number.  I will then show -5 and -7.  They will quickly identify -7 as the furthest from zero when asked. “Is this number the greatest number of the two?”  They will see that the opposite is true. When comparing two negative numbers, the greater number is the one closest to zero. 

Absolute Value. Students may have heard about absolute value in sixth grade, but now it is presented in more detail. I will have students draw a number line and plot two points on it; say -4 and 4. I may also other pairs, using different colors.  I will ask if -4 and 4 are the same distance from zero, and similarly for the other pairs.  These numbers are called opposite numbers.  I will ask them then to give me some opposites: What is the opposite number of 100? of -50? of 25? 

I will then pose a situation in which we say that the numbers represent a football team’s loss on a play, and positive numbers a gain on a play.  The point zero represents the point at which the play originated.   

A loss of 10 yards on a play is represented as -10. Looking now at the opposite pairs I have on the board, I will say that these represent losses and gains on a play. Although a loss of 4 yards can be represented as -4, the distance itself is 4 yards—the distance from the starting point, expressed as a positive number. To make it plainer, suppose someone is wearing a fit-bit and paces out the 4 yard loss. How many yards will show up on the fit-bit?  Will it be -4?  No, it will be a positive number.

Distance is always a positive number. We call the distance from zero on a number line (or, in general, the distance from the starting point) the “absolute value” of a number indicated by two vertical bars; e.g.,   So whether a number is to the left or right of zero, their distance from zero is always expressed as a positive amount. I have them do some examples at this point, mixing in yesterday’s discussion about how to determine whether a number is greater or lesser than another, with today’s discussion of absolute value:

Examples: Find the opposite number:  -3, 5, -500,

Find the greater number:  -3, |-6|; -2, -10; -2, |-2|; |-54|, |53|

This lesson provides a lead in to the next day’s discussion on net gains and losses.

2.  Gains and Losses

This lesson introduces students to the concept of adding negative integers. The approach for this lesson comes from JUMP Math and is a very effective way of introducing students to the concept of adding negative integers through the concept of gains and losses—without students realizing that that’s what they’re doing. After the first few minutes of working with the problems, it is amazing to see them doing intuitively what they will be doing when learning the formal rules for adding negative integers.

When I first started using the technique from JUMP, I mistakenly thought “This is going to be easier than I thought.” As it turns out, while it did make things easier, it wasn’t a slam dunk. Students will still get confused, and will look at what they learned with this technique as something that happened in the distant past and no longer applies. Which is why this particular lesson must be continually brought into subsequent lessons as a reminder that what they did intuitively is what they will continue to be doing. It’s just that the formal rules appear as a different entity.

Opposite Integers. We start with a review of opposite integers. A loss of 5 pounds is represented as -5, and a gain as +5, or just 5. Does a person who loses 5 pounds and then gains 5 pounds end up weighing more, less or the same as their starting weight? Students will generally agree that they will end up with the starting weight. 

Money is an easy and effective way to work with gains and losses. I start by having  students say what the integer is that represents a gain or loss and then state what the opposite integer is. For example, the integer representing a gain of $6 is +6 (or simply 6), and its opposite is -6. I note that while the initial number has the dollar sign attached, when it is expressed as an integer it is without the unit.  A loss of $7 is -7, and its opposite is +7, or 7. (Later in the lesson they will be instructed to write positive numbers without the plus sign.)

Identifying overall gains or losses. Writing +7 – 4 on the board, and explaining that the numbers represent money, I ask was more gained or lost? If there is a net gain, we call it a good day; a net loss is a bad day. They will see immediately that it was a good day. When asked how much was gained, students are quick to tell me $3.

Writing -4 + 4, I ask was anything gained or lost? Nothing was gained or lost, so zero represents “no change”.

We continue with examples. For each one I ask how they came up with the answer. For the problem -6 + 2, there is a loss of $4, (written as -4), which they will explain they derived by subtracting two from six to get a loss of 4.

I paraphrase what they’ve done:

“You have more of a loss than a gain. That means you have more negative numbers than positives. So we write it as an everyday subtraction problem with the signs reversed: 6 – 2. We get positive 4, but since it is a loss, we write it as -4.” 

Nothing fancier than that for now. Students will operate intuitively with the exercises in this lesson.

I point out again that if the first number is positive, like +7-8, we don’t have to write the positive sign.  If we write 7 – 8, it is assumed that the 7 is +7, and is positive. Additional examples help get them used to this, although it may take longer than you would like before there will still be no more blank stares when the “+” sign is omitted from the first number.

Two gains or two losses.  I will write on the board -5 -3=__? and ask if it is a good day or a bad day. If the response is stunned silence, I will state the problem as “I lost $5 and then I lost $3. Good or bad day?”  They will immediately see it is “bad”. I’ll then ask for the overall loss while keeping my fingers crossed that they give me the right answer. They usually do. It is an overall loss of $8, so -8 would be what is written in the blank.

After writing +2+2 =__ students see that the overall gain is $4, so 4 (or +4) is written in the blank. This will be revisited in the next day’s lesson and stated as a rule, that adding two positive numbers results in a positive number, and adding two negative numbers, results in a negative number.

A common error is to interpret two losses of the same number as zero; i.e., – 7 – 7 is mistakenly thought of as 7 – 7.  This mistake will come up repeatedly, and the remedy that I have taken is to remind students of what it represents in terms of two losses: “I lost $7 and I lost $7 more; how much did I lose in all?”

Adding more than two gains and losses.  Now we up the ante a bit, with a problem like +3-4-5. 

This can be solved sequentially. That is, the first two numbers are evaluated: +3-4 which students will know is  -1. Then we are left with -1-5 which students will know is -6.  An easier way is to add the total gains, then add the total losses. In this way we get +3 -9. It is an overall loss of 6 or -6.

Other examples: 2-5-4+8-3.  Adding the gains (and remembering that 2 is the same as +2), we get 2 +8; the losses are -5-4-3.  Total gains equal 10, and total losses are 12, so we have 10-12, for a total loss of 2, or -2. 

For homework I assign problems from JUMP Math (see Figure 1). Problems can also easily be constructed as a worksheet.

Figure 1: Gain/Loss problems from JUMP Math from AP workbook 7.1, Common Core edition, 2015; Toronto; (printed with permission)

  1. Adding Negative Integers on a Number Line

The next day’s lesson now represents what we did with gains and losses on the number line. Also, no matter how much they were on track during the previous day’s lesson, that was yesterday, and today is entirely different.

Because this is a new representation, students may think that the number line method is to be used for some problems, and gains and losses for another.

 It is therefore important to tell students that the number line method is a way to look at what was happening in the previous lesson when we worked with gains and losses. The warm up problems for the day should therefore focus on some of the gains and losses problems, as well as opposite integers:

If a football team gains 8 yards on the first play, how would we write that?  (8)  If they lose 10 yards on the second play, do they have an overall gain or loss?  (loss) By how much? (loss of 2 yards or -2)

The above warm-up, written as a gain/loss problem would be 8-10. I like to use a warm-up problem to segue to the day’s lesson. Using the above problem, students now do the problem on a number line. Rather than writing it as 8-10, however, we write it differently: (+8) + (-10).  The purpose is to emphasize that we are adding a loss of 10 yards.

The rules for showing this on the number line are kept simple: The first number, 8, is marked on the number line. To add a positive number draw an arrow to the right the specified number of units.  To add a negative number we draw an arrow moving left the specified number of units. 

For the above problem we would draw an arrow with a length of 10 units going left from the starting point of 8. It ends at -2 which is the answer, as shown in Figure 2:

Figure 2: Number line representation of (+8) + (-10)

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12.  To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

 ++ = +, and +(-) = -;  Examples:   +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem  as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

Summary of the Rules for Adding Negative Numbers.

  1. The sum of two positive numbers is positive.

     Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

          2.The sum of two negative numbers is negative.

    Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

  1. Adding integers with different signs: informal rule

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive.

Example:  Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.

This is illustrated with an example:

Example:  -5 + 3.  The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2 

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

Common errors. The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

Overthinking and the Lead In to Subtraction. I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day. 

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.

Figure 2: Number line representation of (+8) + (-10)

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12.  To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

 ++ = +, and +(-) = -;  Examples:   +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem  as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

Summary of the Rules for Adding Negative Numbers.

  1. The sum of two positive numbers is positive.

     Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

          2.The sum of two negative numbers is negative.

    Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

  1. Adding integers with different signs: informal rule

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive.

Example:  Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.

This is illustrated with an example:

Example:  -5 + 3.  The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2 

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

Common errors. The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

Overthinking and the Lead In to Subtraction. I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day. 

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.

My Day at Ed Camp

Originally appeared in Education News , Nov, 2015; and is included in “Math Education in the U.S”

I attended an “Ed Camp” recently. This is one of many types of non-professional development and informal gatherings where teachers talk about various education-related topics. The camp I attended was free of charge and took place at a charter school that prided itself on a student-centered approach to learning. In keeping with the school’s focus, the camp also took a student-centered approach which it boasted about in its announcement, calling the event an “unconference”. It stated that the Ed Camp “is not your traditional educational conference; sessions will be created by attendees.”

And that’s exactly what happened. Participants wrote ideas for sessions on Post-It notes which were placed on a whiteboard. The conference organizers then put the Post-It notes in categories that formed various sessions which were then led by whomever wanted to lead them.

The topic suggestions were placed into nine separate categories/sessions. For each of three one-hour periods, there were three sessions that participants could choose from.  I chose “Motivation,” “Feedback in lieu of grades” and “The balance between student-centered and teacher-centered in a classroom.”

A few decades ago there was a mix of opinions on what are considered “best practices” in teaching—some of which included traditional methods. The older generation of teachers, however, has been almost entirely replaced by the new guard. This has resulted in a prevalent new group-think which holds that traditional teaching is outmoded and ineffective. The participants at Ed Camp were of the new guard; mostly people ranging in age between 20’s and 40’s. A few people were in their 50’s or early 60’s, but were subscribed to the same group-think. From what I could tell, I was the only traditionalist present.

Motivation Session

All participants at this session generally agreed that motivation was important and that if classrooms do not have focus, there is loss of attention. They also agreed that students did well in a structured environment and that set routines and clear expectations were motivators. These two consensus items were uttered with the same somnambulant automaticity with which many say grace before chowing down a meal.

Participants then went to town describing various motivating/engaging activities including having students spell out words using their bodies to shape the letters (though I have forgotten what this had to do with whatever was being taught).  After a few more suggestions, someone pointed out that no matter how engaging the activity, the novelty of it wears off, so you can only do it a few times before students are bored — which, I suppose, leaves teachers with the option of more traditional approaches like a warm-up question and then teaching the class.

The issue of group work came up. Group work ranks high on the group-think spectrum as something worthwhile for all students.  So when a teacher said that group work may be difficult for students who are introverts, the feeling of cognitive dissonance was distinctly present in the room.  But the dissonance was quickly dispelled by the same teacher who brought it up. “Well, think of it this way,” he said. “How many times have you gone to professional development sessions and the leader says ‘Now turn to your neighbor and discuss such and such’ and you go ‘Oh, no! Do I have to?’” General agreement ensued.

“But,” he went on, “You kind of think to yourself, ‘Well, OK, let’s get this over with’ and pretty soon you’re doing it and it isn’t that bad. So I think maybe we just have to get kids to think beyond themselves and just go with it, and they’ll see it isn’t that bad.”

I’m fairly certain most of the attendees had been through—and probably hated—professional development sessions that were group-work oriented.  But if there was any disagreement with what he said, it was not voiced.

There was consensus that students responded well to competition.  Teachers noted that students like to see high scores posted or go for extra credit assignments or questions on tests. Such agreement was surprising given that it goes against the trend of the “everyone is special” movement in which all students win awards or graduating classes have multiple valedictorians. Unless one includes competition as being an integral part of collaboration and working in teams and groups, competition would seem to be its antithesis.

Another unexpected result was related by a second grade teacher who taught at the school where the Ed Camp was held. She had assigned her students to groups and arranged her class in clusters of desks as many classrooms are these days. One day her students asked her, “Can we be in rows facing the front of the classroom?”  She tried to reason with them, explaining that when she had been in school she always had to sit in rows and would have loved the opportunity to sit in groups. They told her that it was easier to be in rows because they wouldn’t have to twist around to see what the teacher was doing at the board.  The students assigned themselves numbers randomly so the teacher could put them in straight rows according to their numbers. Since this was a student-centered decision at a school that valued student-centered activities, the teacher reluctantly went along with what they wanted.

Think Pair Share: Harbinger of Things to Come?

The initial premise of the next session I attended—feedback–was that students should be given guidance rather than interim or even final grades. This is not a new concept, as evidenced by a recent comment I saw on a popular education blog: “When numerical/letter grades are king, real learning is kicked to the curb, along with meaningful assessment.”

Like many educational ideas, this one sounds like it ought to be superior to a system of grading that many have accused of being unfair for years, until you get into the details—things like subjectivity and how students will be assessed. The moderator—who did most of the talking in this particular session—said that in guidance-based regimes, students should be told whether they are doing a task correctly or incorrectly and that the key to completing a task was to ensure that students had an appropriate process. I couldn’t be absolutely sure, but it sounded like process trumped content.

He brought up math as an example and said, “I like to give kids problems they don’t know how to do.” This is not the first time I’ve heard this. While I agree that students should be given challenging problems, I also believe that they need to start from a place that they know and advance bit by bit to variants on a basic problem structure to be able to take on non-routine problems.

Such process is known as scaffolding, but modern purveyors of education theory hold that scaffolding should not be used and that flexible thinking –applying prior knowledge to a new and unfamiliar problems or situations—comes with repeated exposure to such problems. Supposedly this develops a “problem solving schema” and “habit of mind” that is independent of acquired procedural skills or facts. But to pull off what this teacher wanted—having them solve something totally different than what they’ve seen—students are given feedback. The feedback is in the form of questions to motivate them to learn what they need to know and ultimately to solve the problem in a “just in time” basis.

The notion of supplying feedback in the form of guidance seemed to this moderator to be a new and cutting edge thing, and in fact announced that the activity of “Think-Pair-Share” was antiquated and should be abandoned. “Think-Pair-Share” has been around for at least 10 years.  The first time I heard about “Think, Pair, Share” was in a course I took in ed school. Briefly, students work together to solve a problem or answer a question, discuss the question with their partner(s) and share their ideas and/or contrasting opinions with the rest of the class.

But now it was considered passé, the main problem with it being that students didn’t know what to say to each other about whatever it was they were to discuss. And that was likely because they had little or no knowledge of the subject that they were supposed to talk about, and which was supposed to give them the insights and knowledge that they previously lacked.

Did this mean that perhaps there was now some evidence that direct and explicit instruction could have beneficial educational outcomes? No. Feedback and guidance was the new “Think Pair Share.” Student-centered and inquiry-based approaches are still alive and well. And in closing, the moderator added that students need good solid relationships with one another and with the teacher. To this end, the moderator said, putting students in straight rows will not build such relationships.

I was tempted to bring up the story of the second grade class that insisted they wanted to be in rows but we were out of time. In fact, we ran over and I was late to the last session on the balance between teacher and student in a student-centered classroom.

Defining Balance—or Not

The conversation in the third and last session of the day was already underway with some talk going on about how effective student-centered communication is fostered using something called “Sentence frames” or “word moves”. These are a set of certain phrases students are encouraged to use when engaging in dialogue, such as, “One point that was not clear to me was ___”, “I see your point but what about ___”, “I’m still not convinced that ___”.

The discussion was in the context of procedures used in conducting student-centered classes. I didn’t know how much about balance they had discussed, and although it is not my habit to interrupt a discussion, I did inject myself using the following sentence frame: “So what do you think is the balance between teacher-centered and student-centered instruction?”

The responses I received were immediate:

“Oh, I just talk at the students forever and go on and on,” said a youngish woman. Another teacher chimed in, “Yes, I tell them that it would be so much easier for them if they just listened…”  This went on for another few seconds, and though I was tempted to use a sentence frame like, “I see your point but what about___?” the one I chose was a bit more aggressive.  “Is that your answer to my question?” I asked. “You think a teacher-centered classroom is all about lecturing with no room for questions or dialogue?”

The woman who first answered me said, “No, I was just being funny.” The conversation turned serious once again with the answer to my question being that the teacher-centered portion of a student-centered classroom is, “teaching the students to be student-centered successfully.” That, roughly translated, means giving them instructions and guidance to do their student-centered inquiry-based assignment.

Example: “In ten minutes, you will complete an outline of what you are going to investigate. Go.”  Ten minutes pass, teacher spot checks various outlines. “Now one person will be the lead investigator, another will be the note-taker, the third person will write the conclusion and the fourth person will do the presentation.”  And so on.

The conversation turned to “student outcomes” and “growth-mindset.” This last phrase, a concept made popular by Carol Dweck, is the theory that students can develop their abilities by believing that they can do so. The term has taken hold as its own motivational poster in classrooms, professional development seminars and Ed Camps across America.  Someone remarked that the idea of growth mindset itself is a student-centered concept. I suppose it is, if you combine belief in yourself with hard work, instruction, and practice—things I don’t hear much about when I hear about growth-mindset.

“Growth-mindset” led into students’ beliefs in themselves, which led to how grades are bad and rubrics were better. A middle school social studies teacher lamented that he was stuck giving students grades because the school district required them, though most of the teachers in his schools used rubrics not to grade, but to provide feedback to students.  (The charter school at which this Ed Camp was held did not give grades, but rather student reports. After the social studies teacher’s lamentation about grades, one teacher who taught at this school cackled “I’m so glad I don’t have to keep grade books anymore!”)

The social studies teacher said that what used to be an A under the old grading system was now a C in his class using his rubric. He didn’t go into details about his rubric except to say that he bases grades on it, and “meets expectations” would be a C.  “I tell parents that I have no problem with a student who gets a C in my class, because that means he or she was meeting expectations. If a student wants better than a C, they can go over the rubric with me to see what is required.”

This struck me as strange. If you give tests and assignments that cover the material and take some effort to do well on them, then maintaining an average of a 90% or more would assure some mastery of the material.  Or does he consider that to be “middle school stuff” and to get an A under his rubric now requires—what?  I never found out. Classes I’ve seen that use rubrics have several: rubrics for group work, presentations, collaboration, essay analysis, presentations and so forth, and there are many categories – like this one for a project presentation in a middle school social studies class . How does one differentiate between “strong student creativity” and “exceptional degree of student creativity” under the “Originality” category? I suspect it’s a matter of “I’ll know it when I see it”.

As time grew shorter, discussions cascaded onto each other, culminating in a discussion about homework. The social studies teacher said he didn’t assign homework, and this turned out to be the practice of most of the teachers in the room. Some of the teachers did report that they received pressure from parents about lack of homework. Parents who ask their kids what they do in school and get the usual “Not much” often follow with “Well, what’s your homework?” and were dismayed to find that the student had none. Parents confronted various teachers, arguing that not assigning homework will not prepare students for the real world. The social studies teacher who was emerging as de facto opinion leader for the session said that in the real world you didn’t have homework, so why should we expect it of our students? This was a bit confusing given that teachers do a lot of work at home. In fact, in many professions it is not unusual to have to do work at home.

But he went on. “And if the real world is high school and college, first of all, not all students go to college. And show me the evidence that homework in high school prepares them for college.” This is the type of argument that seems beguiling if you practice saying it in front of a mirror with an audience applause track playing in your mind. Or alternatively, saying it at Ed Camp sessions like these.

“It is not preparation for the real world,” he repeated, and then clarified that he viewed homework as largely drill and practice activities which in his view held absolutely no value, and certainly, in his opinion, is not something done in the real world. (I should note that I was the only math teacher in this session, but I decided to keep quiet given the reaction when I asked my question at the beginning of the session.)

With parents spotlighted as detractors from how teachers conducted their student-centered classrooms, the session ended with one teacher lamenting how one parent complained that, “This education of my child is becoming my job.”  The teachers all identified with having heard that before. “Gee, sorry to hear that being a parent is so tough” was the general response in the room.

Having been in the position of a parent raising a daughter subjected to student-centered classrooms, I think what that parent meant was not so much, “Why should I be involved in my child’s education?” but rather: “I’m doing a lot of teaching at home that should be going on in the school.” Many parents have complained that students are not being taught grammar, math facts, and other necessities of education, but which teachers of student-centered classrooms consider “drill and kill” and “drudge work.”  That may account for the popularity of learning centers like Sylvan, Huntington and Kumon, which all focus on these things.

The Group-Think of Teaching

Driving home from the Ed Camp, I was reminded of a movie I saw long ago called “The Wicker Man,” in which a deeply Christian, Scottish police officer investigates a missing child on an island in Scotland that practices paganism—and in the end is burnt to death as a human sacrifice to the islanders’ gods. A key point of the film was that the officer’s religion counted for nothing in the midst of different and prevailing beliefs. The winners in such conflicts are those who by virtue of numbers have the means to enforce their beliefs.

I wondered whether in ten years’ time more parents would accept the inquiry-based and student-centered approach more readily as a result of having been subjected to such techniques themselves? Or would there now be a permanent split: parents who came through the system who are happy with their kids being taught as they had been, and parents who had benefitted from the more traditional techniques used in learning centers or from the dwindling number of schools who practiced them?  Would the ideas and techniques discussed at Ed Camp be viewed as outmoded, just as “Think-Pair-Share,” so popular a few years ago, had fallen out of favor? Or would they be replaced by a slight variation of the same thing? Whatever the outcome, it was fairly clear to me that any new educational techniques would be portrayed as a measured and informed decision, a step in the right direction and, of course, progress

Traditional Math (3): Scope and Sequence of Topics for Seventh Grade Math

This is part 3 of a series that will eventually be a book by the same name as this blog. While this is labeled Part 3, it will likely occur much later in the book, after presentation of topics for lower grades (K-6).

The math textbooks in use today have a dearth of good explanations, as well as word problems. Examples usually take the place of any kind of extended explanation, and while explicit instruction relies on worked examples, a textbook should have some additional explanation to go along with them. Indeed, the explanations provided in the teachers’ manuals for these books are instructive and should be included in the students’ textbooks as well.  In addition, today’s textbooks generally include too many topics in one lesson, and the problems at the end of the lesson sometimes go beyond what was discussed in the lesson itself.

Compounding these difficulties are that many of these books tout a “balanced approach”. This usually means that preceding the lessons so described above there is an activity or “inquiry” for students in which they are to discover the general principles behind whatever the topic might be. For example, preceding a lesson on multiplying negative numbers, there may be thought problems on how much weight you would lose if you lost four pounds per week for five weeks, leading to the multiplication of -4 by 5, resulting in a loss of twenty pounds, or -20. The next day, they then have the more straightforward lesson made up mostly of more examples with the rules of multiplication of negative numbers stated formally.

While it is sometimes worthwhile to set aside a day to front load key concepts prior to a lesson, many times these inquiries can and should be incorporated in the main lesson. In a later chapter I will discuss the best way to proceed through such “balanced approach” textbooks.  This chapter focuses on the sequence of topics that I and other teachers with whom I’ve worked believe make the most mathematical and logical sense.

This chapter also discusses textbooks and materials with which to supplement the textbooks that schools have adopted. These additional resources can be obtained from the internet and provide a source of better explanations as well as problems.

What order/sequence makes the most sense?

In one school where I taught, a sixth grade teacher who was hired at the same time as I noted that the math textbooks start off with ratios and proportions. She felt that this might be confusing to students. I noted that the seventh grade textbook did the same thing, but that I had reordered the sequence of topics to make more sense. I felt that starting with integers, and how to operate with negative numbers and then going into rational numbers would be a better way to start. We both agreed that students need some grounding in fractions to better understand ratios.

I’ve noticed many math books for seventh grade math begin with ratios and proportions including the US edition of JUMP Math. I believe this order is followed because it is also the order of topics in the Common Core Math Standards for that grade. There is nothing in the standards that prescribes that order, and I feel certain that teachers will not be terminated for using a different sequence than what appears in the Common Core standards, or in most textbooks that are aligned with the Common Core.

Similarly, I feel that an introduction to algebraic expressions and equations is necessary prior to ratios and proportions. Since ratios and proportions are one of the largest topics in seventh grade math, students need the appropriate background and tools with which to understand the concepts and operate with them to solve problems.

Scope and Sequence for Math 7

What follows is the scope and sequence for a regular seventh grade math course (Math 7), and an accelerated one. For the latter, the additional topics are introduced in a regular eighth grade math course (i.e, Math 8, not Algebra 1).

Key topics for which this series provides descriptions of their presentation in the traditional manner are: Integers, Rational Numbers, Expressions and Equations, Inequalities, Ratios and Proportions, and Percents

I. Integers

A. Integers and absolute value

B. Negative numbers (Adding, subtracting, multiplying and dividing)

II. Rational numbers

A. Rational numbers (general definition; ordering)

B. Adding rational numbers

C. Subtracting rational numbers

D. Multiplying and dividing rational numbers

III. Expressions and Equations

A. Algebraic expressions

B. Adding and subtracting linear expressions

C. Solving equations using addition or subtraction

D. Solving equations using multiplication or division

E. Solving two-step equations

F. Multi-step equations (as necessary; recommended for accelerated classes)

G. Word problems

IV. Inequalities

A. Writing and graphing inequalities

B. Solving inequalities using addition or subtraction

C. Solving inequalities using multiplication or division

D. Solving two-step inequalities

E. Word problems

V. Ratios and proportions

A. Ratios and rates

B. Proportions

C. Writing proportions

D. Solving proportions

E. Slope

F. Direct variation

VI.  Percents

A. Percents and decimals

B. Comparing and ordering fractions, decimals, and percents

C. The percent proportion

D. The percent equation

E. Percents of change: increase and decrease

F. Discounts and markups

G. Simple interest

H. Compound interest (for accelerated classes)

VII. Constructions and scale drawings

A. Adjacent and vertical angles

B. Complementary and supplementary angles

C. Triangles

D. Quadrilaterals

E. Scale drawings

VIII. Area

A. Review of area of triangles and quadrilaterals

B. Circles and circumference

C. Perimeters of composite figures

D. Areas of circles

E. Areas of composite figures

IX. Surface area and volume

A. Surface areas of prisms

B. Surface areas of pyramids

C. Surface areas of cylinders

D. Volumes of prisms

E. Volumes of pyramids

X. Probability and Statistics

A. Outcomes and events

B. Probability

C. Experimental and theoretical probability

D. Permutations

E. Combinations (for accelerated classes)

F. Compound events

E. Independent and dependent events

F. Samples and populations

G. Mean, Median and Mode

H. Comparing populations

Extra topics for accelerated classes (these are covered in Math 8)

XI.  Transformation (Translations, Reflections, Rotations)

XII.  Graphing and writing linear equations

XIII. Real Numbers and the Pythagorean Theorem

XIV. Volumes of Cylinders, Cones and Spheres

XV. Exponents

Additional Resources with Which to Supplement Textbooks

JUMP Math

Singapore’s Primary Math Series (U.S. edition)

Pre-Algebra: An Accelerated Course (Dolciani, Sorgenfrey, Graham); Houghton Mifflin Company. 1988.

Traditional Math (2): Prefatory Remarks, Disclaimers and Warnings

This is part 2 of a series that will eventually be a book by the same name as this blog. Enormous changes to what you read here will likely happen at the last minute before publication, so enjoy this raw version while you can.

I believe strongly in how math should be taught and even more strongly in how it should not be taught. Nevertheless, when I am involved in teaching it as I believe it should be taught, (which happens to be in the traditional mode) I feel vaguely guilty, as if I am doing something against the rules and perhaps even wrong. 

I have heard from other teachers who identify and empathize with this. Traditional math teaching is vilified by education schools, education consultants and other educational rent-seekers that have pervaded in the profession over the last three decades. Teachers feel bad for teaching using a method that has been proven to be effective. Practicing math problems is disparaged as “drill and kill”.  Whole class instruction is thought to be ineffective because it doesn’t promote collaboration.

The picture that many have when hearing the term “traditional math” is a classroom in which seats are arranged in straight rows, the teacher stands at the front of the room and lectures non-stop for the duration of the class, students learn all procedures and problem solving methods by rote, and no background on the conceptual underpinnings of same are presented. Word problems are dull and uninteresting and students do not feel any desire to try and solve them. Problem sets (commonly called “practice problems”) are repetitious and do not present any challenge.

The traditional mode of teaching math has been slowly and steadily displaced over the past three decades by reform methods and are seen mostly in the lower grades. High school has remained somewhat impervious to this displacement; middle school less so. The methods for teaching math in the traditional manner is rarely if ever taught in schools of education. The result has been that newly arriving teachers from ed schools have been steeped in the math reform methods and are taught that inquiry- and activity-based student-centered, collaborative teaching is superior to explicit and whole class instruction.

Defenders of the reform methods will insist that teachers use both explicit and inquiry methods, and therefore teachers on both sides of the aisle are saying the same thing.  We are not.  Yes, some amount of inquiry and other methods co-exist within explicit instruction as are activities and group work.  Just not to the same degree.

In this series we will therefore discuss what it is we are explicit about when we teach math explicitly. What is it that we say in terms of explanation, and what questions do we ask of our students?  With that, let’s look at what goes into a typical traditional approach in a classroom.

Traditional Classroom Approach

The following is a typical daily approach in traditional math teaching. There are other aspects of traditional teaching that in the interest of brevity and staying to the topic of math education, I don’t go into here. If you are interested in the general principles of traditional instruction, however, you may wish to read Tom Sherrington’s book “Rosenshine’s Principles in Action”, which describe Barak Rosenshine’s principles of instruction.   

Warm-ups: These are four or five problems that students work on in the first five to ten minutes of class.  Some of the problems are from previous lessons to keep old material fresh. Others are what has just been covered. And still others may be problems that lead into the day’s lesson. For example, if a class has learned how to factor trinomials in which all the terms are positive, such as x^2 + 5x + 6 there might be a problem where they are asked to factor x^2+x – 6 some students can make the leap; others have questions for which I provide hints. When it comes time to go over the warm-ups, this last problem will then set the stage for the lesson to come which focuses on trinomials where the signs are not all positive.

Go Over Homework Problems:  I have provided answers to homework from the previous day so students have checked their work. Therefore, I spend this time going over problems that they find difficult—usually three or four. I may have a student who has done the problem correctly explain it; otherwise I explain.

The Lesson and Start on Homework: I then go into the lesson. This series will go into what is talked about explicitly, which includes the worked examples that are part and parcel to the instruction. The pattern followed is the “I do, we do, you do” technique, in which students are given problems to solve after a few initial worked examples that are done together.

I leave enough time (approximately 15 minutes) at the end of the lesson for students to start working on their assigned problems. This allows me to answer questions and provide help. This helps prevent the situation of students not knowing how to do the problems because they may have forgotten how to proceed. Starting the homework problems in class allows for practice and the learning that comes from it.

On the topic of worked examples, a paper by Liljedahl and Allan (2013) sheds some light on the topic that I believe is useful to understanding where I stand. It talks about what they term “Now try this one” problems and states “These are the problems assigned, usually one at a time, by a classroom teacher immediately after s/he has done some direct instruction concluding with some worked examples. We recognize the rather traditional approach in this method of teaching and, although we would not ourselves approach the teaching of the topics in this fashion, we make no judgement about it here.”

Although deferring judgment one can guess how Liljedahl and Allan really feel about traditional classrooms. This is made a bit clearer by Pershan (2021) who doesn’t hold back in his book “Teaching Math with Examples”. He states: “Some of the dullest teaching on the planet comes courtesy of worked example abusers. These are the math classes that consist of a steady march of definitions, explanations and examples, one after the next. Practice (and learning) happen out of the classroom hours later, while students work on their homework.”

As I said, I start students on their homework in class, not hours later, and as you will see in this series, the explicit instruction is not a steady march of definitions and explanations. There are questions, and—dare I say it—even some inquiry along the way. But as far as examples, with some exceptions discussed below, Liljedahl, Allan and Pershan have pretty much nailed my traditional “Now try this one” approach.

For those who find such approach offensive, you may stop reading at this point. For others who are curious, I will say that my traditional approach does not quite fit the definition of the “worked example abuser” much as some would like to believe, nor is it worthy of what I imagine is the negative judgment that Liljedahl and Allan have deferred. I offer problems that are scaffolded and ramped up in complexity and difficulty so that there is in fact learning involved in doing the examples as you will see if you stick with this series.

I will say here that Pershan’s book does in fact offer good advice and methods for presenting examples and is worth reading. I would also add that this series on traditional math teaching is also worth your while and which I hope will dispel mischaracterizations and myths about traditional math.

References

Liljedahl, Peter and D. Allan (2013). In Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1., Kiel, Germany: PME.

Pershan (2021).  Teaching Math with Examples.  John Catt Educational, Ltd., Woodbridge UK; Clearwater, Florida

A new series: Teaching traditional math

I believe strongly in how math should be taught and even more strongly in how it should not be taught. Nevertheless, when I am involved in teaching it as I believe it should be taught, (which happens to be in the traditional mode) I feel vaguely guilty, as if I am doing something against the rules and perhaps even wrong. 

I have heard from other teachers who identify and empathize with this. Traditional math teaching is vilified by educationists. Teachers feel bad for teaching using a method that has been proven to be effective. Practicing math problems is disparaged as “drill and kill”.  Whole class instruction is thought to be ineffective because it doesn’t promote collaboration. The picture that many have when hearing the term “traditional math” is a classroom in which seats are arranged in straight rows, the teacher stands at the front of the room and lectures non-stop for the duration of the class, students learn all procedures and problem solving methods by rote, and no background on the conceptual underpinnings of same are presented. Topics are presented in isolated fashion with no connections with any other topics, so that students are prevented from seeing how one mathematical idea may relate to another. Word problems are dull and uninteresting and students do not feel any desire to try and solve them. They have no bearing on any aspect of students’ lives, and all information needed to solve the problem are contained within the problem itself.  Problem sets (commonly called “practice problems”) are repetitious and do not present any challenge.

I could go on, but I’m sure you’ve heard the many ways that traditional math is mischaracterized. And yet, despite the slings and arrows that are hurled toward traditional math, there are teachers who continue to teach math in a traditional manner. Traditional math teaching incorporates pedagogical methods that have been proven to be effective, like direct/explicit instruction, worked examples, and scaffolded problems and, despite the claims that such methods have failed thousands of students, has produced successfully achieving students.

This series is not about the pedagogy of traditional math. Rather, it provides examples of how math is taught in the traditional manner. It discusses the type of worked examples used, and how previously covered topics are kept fresh so that students remember the procedures when they come up again—and they do come up again. In all the topics covered, it makes explicit what is meant by explicit/direct instruction; that is, what is made explicit for each of the key topics. It also will show how some amount of inquiry and activity is part and parcel to explicit teaching.in the process. Most importantly, it shows not only how procedures are taught, but the conceptual understanding behind it while recognizing what levels of understanding students will likely retain–without obsessing over it.

The series has two purposes; 1) For approaches that are similar to what you already do, it may give you the assurance that you are not the only one who does these things and that you are not crazy, and 2) It may give you some new ideas.

The series is focused on key topics covered in K-6, as well as seventh and eighth grades (the latter including the regular Math 8 as well as eighth grade algebra). The foundational aspects of math in the lower grades will be discussed with reference to books such as Singapore’s and other useful series. For all grades, the book will also provide a list of books one can obtain from the internet or other sources that can supplement the textbooks that a school has adopted. Which brings me to the topic of textbooks.

Textbooks

Textbooks are handy things to have because they contain a sequence of topics and a breakdown of what gets taught within each one. Unfortunately, many textbooks are poorly written. There is very little explanation of how a procedure works, and frequently two or three sub-topics are embedded in a single lesson. For example, in one algebra book I saw, there were two types of word problems presented in one single lesson: 1) Mixture problems, such as: “How many liters each of 20% and 50% sulfuric acid must be mixed to obtain 30 liters of a 45% sulfuric acid solution” ; and 2) Wind and current problems such as “A plane flies with the wind for 2 hours and travels 360 miles; when flying against the wind, it takes 3 hours to cover the same distance. What is the speed of the plane in still air, and the speed of the wind?” While the problems are set up similarly when solving with two variables, it is a lot of information to present in one lesson. I would break it up into two separate lessons.

The approach in this series (which will ultimately become a book) is to provide workarounds to the various shortcomings of textbooks. This frequently involves supplementing the textbook with material in other books, which I will recommend. For seventh grade and regular textbooks I have usually had to reorder the sequence of topics for a more logical flow, add some material within each topic area, and provide problems from other books. It does not involve designing a new curriculum from scratch. For eighth grade algebra, my solution has been to use a 1962 algebra book by Dolciani. While this book is very good, its availability on the internet has decreased to the point that they are now very expensive. There are other books that may be used, however and I will provide a listing.

For all courses, the series will describe how to address some of the topics that are included in Common Core, but which, in my opinion and others with whom I’ve consulted on such matters, do not need the degree of emphasis which is typically given in textbooks. For example, in seventh grade, proportions can be presented simply as they have in the past. Students should be given the opportunity to solve many types of problems using proportions. They do not need at this point to spend a lot of time identifying the constant of proportionality (or variation) or expressing the proportional relationship as an equation. They will do this in algebra where they learn about direct variation and how to represent such relationships as equations. It will make more sense then because they will have mastered the foundational aspects of proportions. i.e., they learn about direct variation, and how to represent directly proportional relationships as equations.

I recognize, however, that end of year state testing may include questions on this and other topics, so I am careful to explain it, show how it is done, and put a question or two on the quiz or test and give extra credit points for those students who can do it.  In short, it is covered but not obsessed over.

A quick example of how this series will unfold

As an example of how this series will proceed, here is a discussion about how to handle the situation when a student gives the wrong answer in front of the class.

I frequently hear that it is not a good idea to tell a student that they are wrong, even though some advocates say that making mistakes should be a goal of teaching math. It is not a goal for me, but mistakes will happen. I don’t shy away from telling a student that they are incorrect. Some people use mini-whiteboards that students write their answers on, and then hold them up. 

My method, when a student says the wrong answer, is to say, “Not what I got”. If I can see what the mistake was I will sometimes say “Oh, it looks like you multiplied instead of divided”, or whatever the mistake happens to be. Or I may ask the student to show how he or she obtained their answer which provides insight into what the mistake was, leading to how to do it right. I might go around the room. If there are many mistakes, making a game of it, writing down the answers on the board until the correct answer comes up. These methods have worked well, and I haven’t seen students become unduly depressed with such approaches.

Another method, usually when introducing a new topic is to preface the problem with “I am willing to bet $100 that the answer you give me when you hear this problem is going to be wrong.” This serves as a dare, and students will rise to the challenge. One problem that I use when introducing algebraic equations to solve word problems is: “John and his sister have $110 between them. John has $100 more than his sister.  How much money does each person have?”

Usually the first answer I hear is $100 and $10.  I show quickly why this cannot be right. Students then resort to guess and check and finally hit upon the right answer: $105 and $5.  This serves as a segue to how using an equation is a much more efficient way to get the answer—something I’ll get into more in a later chapter. It also serves as a way for them to say an answer out loud without feeling embarrassed if it is wrong.

Next chapter: Early grades and what should be covered.

For an in-class look at traditional math, check out “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”.

Prevention Equals Treatment, Dept.

NOTE: For those interested in math education issues, the Science of Math organization has been formed to do for math education what valid research has done for the science of reading. Consider becoming an affiliate of Science of Math to support the efforts of this organization.

A recent paper by Sweller discusses how and why inquiry based approaches harm student learning and is largely ineffective. This is not the first paper that Sweller has written about the subject (not to mention those written by Kirschner), but it may be the most definitive so far.  The paper has sparked interesting observations. In particular, I was intrigued by a friend’s remarks that the prevailing mindset in schools and districts is that only students with learning disabilities need direct instruction and that it functions primarily as a student services support.

In math education circles direct (or explicit) instruction has been painted as the prevalent currency of traditionally taught math. And those who seek to reform math education tend to deride and mischaracterize traditionally taught math as  1) consisting solely of direct/explicit instruction with no engaging questions or challenging problems, 2) focusing on rote memorization and no conceptual understanding, and 3) failing to teach math in any complexity. In fact, traditionally taught math employs some inquiry based approaches, while reform math teaching generally relies more on discovery/inquiry approaches than direct. (A paper by Anna Stokke (2015) addresses what an appropriate balance of direct and inquiry-based instruction should be and states: “One way to redress the balance between instructional techniques that are effective and those that are less so would be to follow an 80/20 rule whereby at least 80 percent of instructional time is devoted to direct instructional techniques and 20 percent of instructional time (at most) favours discovery-based techniques.” This was corroborated in a paper by Adam Jang-Jones (2019) which quantified the “sweet spot” between inquiry and discovery based approaches.)

Reform approaches in math in the lower grades (K to 6) have steadily grown over the past three decades. I had long wondered whether some diagnoses of math learning disabilities (MD) of students were in fact incidents of low achievement (LA) due to lack of access to effective instruction. In other words, did the inadequacies of reform math mimic cognitive deficits?

I was therefore surprised and delighted to learn, when I took an Introduction to Special Ed class in ed school, that there was no beating about the bush when talking about students with learning disabilities and other disorders. We learned that students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction along with other methods. This was mentioned in our textbook (Rosenberg, et al., 2008) and often repeated by our teacher—a tremendously kind teacher named Carmen. (Interestingly, this course was one that was not required as part of my certificate program.)

Of particular interest in this course was the topic of “Response to Intervention” (RtI). RtI is a procedure in which struggling students are pulled out of class and given alternative instruction which includes direct instruction and other evidence-based approaches supported by randomized control trial studies. If they improve under RtI, then the student is presumed to not have a learning disability and is returned to the normal class. If they do not improve, that is an indication that they have an underlying learning disability. (The procedure was established under the Individuals with Disabilities Education Improvement Act (IDEIA) passed in 2004).

I recall the discussion we had about RtI in which I posed a hypothetical to the teacher. “Suppose someone is doing poorly in a math class that relies on an inquiry-based math program,” I said. “And they pull the student out and give him RtI using direct instruction and other techniques, and this student does well.  What happens next?”

“Then the student is placed back to the class during math.”

“But then suppose the student does poorly again? Wouldn’t that indicate that he needs more direct instruction rather than inquiry based approaches?”

“It doesn’t work that way,” Carmen said.

“So he’s stuck with a program in which he doesn’t do well.”

“Right,” she said.  “But if he did poorly in RtI, then that would be evidence he has a learning disability.”

What Carmen was telling me was the Catch-22 aspect of special ed. That is, in schools that rely on programs that follow math reform principles, approaches used in traditional math teaching are generally not an option unless a student qualifies as being learning disabled. And if under RtI, a student does well with direct instruction this is taken as evidence that the student does not have a learning disability.

I suspect that the use of RtI is higher in schools that rely on reform-based programs. I would like to see research conducted to see if that is true. From where I and many teachers and parents sit, the effective treatment for many students who are LA, is also the effective preventative measure.

Based on conversations I’ve had with education professors, I believe the educational establishment will likely continue to resist recognizing the merits of traditional math teaching and direct instruction. The following statement from James McLesky (2015), one of the authors of the textbook we used in the special ed class and a professor at University of Florida’s College of Education, is typical of what I’ve been told:

If we provide only (or mostly) skills and drills for students with disabilities, or those who are at risk for having disabilities, this is certainly not sufficient. Students need to also have access to a rich curriculum which motivates them to learn reading, math, or whatever the content may be, in all of its complexity. Thus, a blend of systematic, direct instruction and high quality core instruction in the general education classroom seems to be what most students need and benefit from. 

Statements such as these imply that students who respond to a diet of more direct instruction constitute a group who may simply learn better on a superficial level. I fear that RtI will evolve to incorporate some of the pedagogical features of reform math that has resulted in the use of RtI in the first place.

I am hoping that the publication of Sweller’s latest paper and the reaction to it that I’ve seen so far, will result in an increasing recognition of the benefits of direct instruction specifically and traditional instruction generally, as well as the harm that can result from inquiry-based approaches. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.

Reference:

McLesky, James. (2015) Private email to Barry Garelick. November.

Rosenberg, Michael; D. L. Westling, J. McLesky (2008). Special Education for Today’s Teachers: An Introduction. Pearson. New York.

Doing it wrong, Dept.

Talking about teaching math opens one up to choruses of “You’re doing it all wrong” among those who have been indoctrinated into the various catechisms of math education. One of those is “Never tell a student they made a mistake”. I guess this is because it affects their confidence and self-esteem and therefore is anti-growth-mindset. (On the other hand, we have Jo Boaler telling us that students should be encouraged to make mistakes because it makes their brains grow.)

I have no problem telling a student they made a mistake, though I do it by saying “That’s not what I got. Anyone else get an answer?” When many students make a mistake I capitalize on this and say “OK, so far we have …. ” (I rattle off the various answers), and then many hands go up among those who want to be part of what is now perceived as a fun game. But if only one person makes the mistake, I’ll try to see if they know what they did wrong. Sometimes the student knows; other times, I might know and I’ll give my opinion. And still other times we don’t know, but I’ll give another similar problem and the student who made the mistake usually will try again. At least that’s been my experience. But to make the math ed progressives reading this entry feel better, I’m no doubt doing something wrong.

For eighth graders, it’s a little trickier, because they are very self-aware at this stage of their lives and can be very guarded. Some teachers use mini-whiteboards on which students write the answer and hold up the boards for the teacher to then say “Right, right, nope–try again, …” etc. I do a variation of this. I don’t use mini-whiteboards. Instead, I’ll tell them to do the problem in their notebooks, and then I go around. If someone has the wrong answer and they write out their steps, I can point out the mistake, and they can then re-do it. For those who get it right, I’ll tell them so. If the person who got it wrong initially then gets the right answer, I call on that person to tell the answer to the class. In this way, the person is not singled out for making a mistake, and they feel confident in giving the answer to the class, knowing it’s correct and not fearing the teacher saying that it’s wrong in front of their peers.

But when time pressure is an issue, you sometimes have no choice but to tell someone they are wrong. I make note of those who are not getting it, and during the time that I allot for students to start on their homework (a term which has now morphed into “practice problems”–I guess “homework” is too risky a word in view of self-esteem and growth mindset fantasies) I spend the most time working with them.

For those students in eighth grade who really should not be in such class but who are placed there because of parents’ insistence, there are a number of options I exercise. I may recommend to the parents that they hire a tutor. Another alternative (which may occur even if the student has a tutor) is to recommend that the student repeat algebra in 9th grade. Some go along with this, but others do not.

If these ideas are offensive to some of you, please realize that I wear my shirts tucked in, avoid Apple products, and use a point-and-shoot digital camera rather than take pictures with my cell phone. I am semi-anachronistic and am determined to stay that way. It’s only a matter of time before my out-of-date habits become the latest fad. By that time, I’ll likely be dead, in which case they’ll probably name a brick-and-mortar bookstore after me.

The Math Wars Continue, Dept.

I always get a kick (as well as a wave of nausea) when I hear arguments about how math should be taught referred to as “math wars stuff”. Such criticism implies that the we are long past the math wars and that they were just trivial spats that signified nothing. In a communication I once had with Jay Mathews–who for many years has spewed his arrogant views of education in a column he writes for the Washington Post–he said that the math wars were two groups of smart people calling each other names.

I won’t comment on the word “smart” here, other than to say it’s overused to the extent that it means nothing, and has become a code word for edu-pundits who compliment each other by saying so and so “wrote a smart and thoughtful post” about whatever.

Well, I had the opportunity to write a “smart and thoughtful post” on math education, courtesy of Rick Hess who invited me to do so. It was published first at Ed Week, then at AEI, and finally at Education Next’s blog. While it has proved a popular piece, there was a recent take-down of it, also published at Education Next’s blog. The author works for TERC which publishes Investigations in Number, Data and Space, which in my opinion and others whose opinions I respect, is one of the worst of the NSF-sponsored atrocities.

I was about to defend my stance, when to my great relief, Sanjoy Mahajan, a research associate in mathematics at MIT did the heavy lifting for me on Twitter, reproduced below:

  1. There’s so much to say about that clever nonsense. There’s the straw man of “practicing procedures alone” bringing understanding. But Mighton’s new book _All Things Being Equal_, pp. 98-102, has a great treatment of the long-division procedure with understanding.
  2. There’s the sneaking in of “when division applies in solving real-world problems” onto (into?) the list of concepts underlying the long-division ALGORITHM. Sure, it belongs on the list underlying division — but not underlying the algorithm.
  3. After these concepts, mostly valid, comes a call to develop a “multi-faceted view of division.” But I want students to understand not all ways that one could divide but rather the long-division algorithm.
  4. And the method offered will not help: the “rich task” of justifying the algorithm for “any two RATIONAL NUMBERS.”  That choice is either sloppy or insane. I have never used the long-division algorithm for arbitrary fractions, only decimal numbers.
  5. About the incessant calls for “authentic mathematics.” It’s rich coming from educators whose favorite incantation for stopping any rigorous teaching, e.g. long multiplication, is “development [un]readiness.”
  6. Proving this conjecture is not at all authentic mathematics. The main effort of mathematicians, not evident from the format of journal articles, is making the conjectures.
  7. Finally: Even if it were authentic, where’s the argument to show that acting with the outer forms, but without the inner knowledge, of mathematicians makes you understand math like mathematicians? It’s cargo-cult thinking.

I hope you enjoyed this foray into the supposedly defunct math wars.