Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Archive for the ‘remedial/developmental math’ Category

Unequations Buzz

Posted by mark schwartz on August 11, 2016


Had a thought. Simple one-variable 1st degree equations, by definition, state that there is a bunch of stuff “here” that equals a bunch of stuff “there”. For example, 2(3x ̶ 1) = 5(x + 1). What is meant by “equal”? Looking at this equation, obviously the two bunches of stuff are not equal! What this statement means is that if you can find the value of the variable “x”, replace the “x’ with that value in both sides of the equation and evaluate both sides, the value on both sides of the equation sign will be equal. Thus, that’s why one solves for the value of “x”.

The fundamental rule for solving equations is “whatever you do to one side of the equation, you do to the other side.” This, in essence, maintains the equality. My thought was that rather than start with an equality and burp out the rules, start with an unequation and have students play with it to find out how to make it an equation. However, we won’t use paper and pencil; we’ll use poker chips.

The Story

In order to solve an equation of this order, students need to know a lot of stuff – identification of terms, order of operations, distributive law, the four basic operations with signed numbers and to verify their answer, substitution of a value for the unknown and of course the basic rule of “whatever you do to one side of the equation, you do to the other side.”

Solving unequations is simpler and is a kinesthetic, visual way to have students play with all those things which, in my view, expands their conception of equations. In many instances, I’ve seen students who know all the elements but somehow can’t blend them together to solve equations. Here’s how unequations work.

Each group of students (2 or 3 to a group) gets a handful of white poker chips and each chip has a positive on one side and a negative on the other. You can use other markers if you choose.

I ask them to put 1 to 5 chips in each pile but the total value in each pile can’t be the same. Two questions that always comes up are (1) can we put positives in one pile and negatives in the other and (2) can we put positives and negatives in the same pile? So, right away, they’re thinking about this exercise; they’re engaged. We have a discussion about this and although they don’t yet know what to do with these 2 piles (although some guess they’re equations), I let them determine what is allowable. So again, right away they “own” this exercise because they have determined what’s allowable. By the way, the discussion about what is allowable has many branches and typically includes a lot of “what if” banter. I just listen.

Once this is resolved, I then ask them to label the pile on the left “A” and the pile on the right “B”. This also is fun because there typically is someone who stacks the piles vertically rather than horizontally, so I simply say the pile furthest from you is A and the pile closest is B.

When everyone is ready I then ask them to do something to their pile A such that the total value in both piles is equal. This is also a fun point in the exercise for classes that allow positives in one pile and negatives in the other, but overall the buzz within each group again is one of the goals of this exercise. When this is done, I ask them to return to their original piles and then I ask them to do something to their pile B such that the total value in both piles is equal.

In both cases, I ask them if there was only 1 way to make the piles equal. Buzz, buzz again and the consensus was yes.

The next question to them was do something to both piles at the same time such that the total value in both piles is equal. This really generates buzz and questions to me, which I say I’ll answer later. The reason I won’t answer is that I want them to explore how this works. What they discover is that there is an unlimited number of ways to do this. For example, if A = 2 and B = 4, add 5 to A and 3 to B and both piles equal 7. There usually is an “aha” moment when they realize that as long as the difference between the two numbers added to A and B is 2, the total value will always be equal. Some also discover that unequal amounts can be subtracted from both piles and further that two numbers differing by 2 can result in an equal value in both piles. And there’s another “aha” moment – the total value in both piles can be negative if both were positive at first! And what’s more, zero is a valid value!

So, we played with these 3 options for a while and there was discussion all along about not only what was allowable but also the range of answers under the different conditions. Then we moved to equal piles to begin the exercise.

I ask them to adjust their piles so that there is an equal number in both piles. This then brings up the issue of their rule allowing positives in one pile and negatives in the other, if they allowed this. They realize they have to rule it out. But I then ask if they can have an equal value in both piles while having positives and negatives in the same pile. Can the total in both piles be positive or negative? Buzz, buzz and the conclusion is that it’s ok but this comes after a lot of discussion and this really gets them going about signed numbers. For example, if they are to have 3 positives in both piles to begin, they could put 4 positives and 1 negative, or 6 positives and 3 negatives or … here it goes again with an unlimited number of both as long as the total is 3.

So, I ask them to consider there beginning equal value in both piles and typically they make it simple – either all positive or all negative and they do this partly – they tell me – because they don’t know what I’m going to ask them to do. At this point, the equation question arises and I have to admit that we’re headed in that direction. After playing with this for a while, the class concludes (again) that there is an unlimited number of values that can be added or subtracted to maintain the inequality.

The next step is to give each group a few blue chips. What the group is asked to do is have one person look away of shut their eyes while the others in the group do two things: (1) set up two piles with an equal number of chips in both and (2) remove a certain number of chips from one of the piles and place a blue chip in that pile. In essence, create a simple equation. When they are done setting it up, the closed-eye person is to look at what they’ve done and answer the question: what must you replace the blue chip with in order to make the piles have equal value?

Do each of these exercises until the class seems comfortable with all the ideas that got buzzed about.

At this point, if you’d like to extend this 2-pile concept to work with introducing work with equations, see Chipping Away at Equations in this blog. It links up with this posting and together it gives students a different view of equations.


Posted in algebra, basic math operations, equations, math instruction, mathematics, remedial/developmental math | Tagged: , , , , , | Leave a Comment »

Walk the Clock: It’s Fractions

Posted by mark schwartz on August 3, 2016


For some reason, or perhaps reasons, fractions don’t make sense to many students. Despite the visual representations in text and/or the use of manipulatives such as Cuisenaire rods, fractions seem to remain a mystery to students. One day I asked all my basic math classes “What makes fractions so hard?” The overwhelming response focused on remembering the steps of the 4 basic operations. For them, operations with fractions seemed nothing more than trying to remember the steps to get the answer. Somehow, math instruction throughout elementary and secondary education led students to think not about what fractions mean and what they represent but rather to think about how to “solve the problem”. So, I played with an idea which seems to have provided a way for students to “see” fractions a little differently.

The Story

DON’T TELL STUDENTS THIS IS FRACTIONS! If someone asks if this is fractions, tell them it will be discussed after the activities are done. I’ve provided an idea on how to do this in the discussion section following the demonstration of the activity.

The students preferably will work in groups of 3 (or 2, depending on the size of the class). The minimum grouping is 2. Each group gets a magic marker and 12 paper plates. The students are to number the plates 1 to 12. The plates are to be placed on the ground as a clock face. This activity is best done outdoors but if not, move the desks and chairs to allow for each group to have enough space for one person to walk inside and one person to walk outside a clock face circle. If neither of these spaces are available, the plates can be cut down in size and placed on a table top. If it’s a rainy cold day and going outside is a bad idea, and if the curriculum allows and time allows, make it a “review” day and hope for sunny and warm tomorrow. This activity works indoors and on the desk top but outside is best; it’s more fun. If done on the desk top, 2 markers per group will be needed. These markers will be the “walkers” in the activity (this will be explained below).

Here’s how it works. Lay out the plates as a clock face. One of the people in the group will walk outside the circle (call this person the outsider); one person will walk inside the circle (call this person the insider); the third person will be the reader/recorder (call this person reader). Give each group a copy of the activities (a sample is below) which states what the insider, outsider and reader are to do. Once the groups have figured out who will do what, give a demonstration of what they are to do, using the 1st activity.

Using the first activity and using one group to demonstrate, note that both walkers will walk twice. Both walkers start at “12”. In each activity, the insider walks first and then the outsider. The first walk is done when the insider reaches “12”. The second walk for both starts where their first walk ended. The reader is to watch and verify that each walker takes the right number of steps (others in that group can help verify).

1st Activity: on the first walk, the insider walks 2 units while the outsider walks 1 unit. On the second walk, the insider walks 3 units while the outsider walks 1 unit. The reader will note “outsider.insider”. In this activity, the record should show 10.12.

If there is confusion about the walking and/or the recording, just repeat the first activity. When everyone’s ready, move on to the next activities.

2nd Activity: on the first walk, the “insider” walks 4 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 6 units while the “outsider” walks 1 unit. The reader should note 5.12.

3rd Activity: on the first walk, the “insider” walks 2 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 3 units while the “outsider” walks 2 unit. The reader should note 14.12. (There should be questions on how to record this. Show students “military” time.

It’s important that if more activities are to be done, don’t allow students to do it. The reason: activities provided by students may result in a very time consuming set of walks and more critically, present a new issue to handle. For example, although subtraction of fractions can be done this way, I suggest not doing it. You could get a negative answer and you might want to avoid this. Just stick with one concept at a time; adding fractions (although they may not realize it). Given this, you might want to prepare and walk through a bunch of activities and be careful that none of them take too much time, yet enough time for the students to play and enjoy it.

Again, do not say anything about fractions at this point, but what has happened is that the problem 1/2 + 1/3 has been done. The record “outsider.insider” is 10.12 , or in reduced fractional notation is 5/6. Most likely, someone has noticed that the insider always has a value of 12. You sort of have to weasel your way around this and don’t yet call it a common denominator.

A Little Discussion. After these activities, you can transition to presenting fractions as you usually do. But, here’s one idea to consider in talking to students about how this activity demonstrates addition of fractions. What is seen and used but not referenced is the common denominator of 12. This explains why the insider’s walking the line twice isn’t counted twice. In the problem 2/3 + 3/4 , the denominator could be any multiple of 12 but in this case since it is 12 and you know it, don’t count it twice. Students may balk at this idea but it can be explained further. The insider always walks the line twice but always restarts the 2nd walk at “12”, while the outsider restarts the 2nd walk where the first walk ended so the insider’s walks aren’t added, rather they simply repeat.

Also not seen is the addition, but it occurs in the outside walk when the second walk starts where the first walk ended. The outside walker’s position at the end of the first walk is added to the beginning of the second walk. Please note that using this method for a problem such as 1/2 + 1/3 would give the answer 10/12, not 5/6, so clearly reducing fractions has to be addressed before this activity. Further, you might question how to get from this activity to the “rules” for addition and subtraction, but that’s not the point, although it can be seen because both fractions in this example, were converted to equivalent fractions with a denominator of 12, although in this case and others, it wouldn’t necessarily be the lowest common denominator. This again, could create a teaching moment, discussing the issue of common denominator versus lowest common denominator.

I suggest that different sets of students get a chance to walk the line. In fact, teams of students could do it; two walk and the others verify that their walking is accurate. Further, the point at which the transition from this activity to the traditional fraction work is to be made is a matter of how the class is collectively responding. In some instances, students caught on and realized that this was adding fractions. But even if they caught on, I still had them walk through all the activities. In several classes, students wanted more exercises. I think it was because it was a nice warm day. It’s a judgment call.

One more thing. Recall that the purpose of this activity is to give a visual and kinesthetic sense to the “rules” and it does seem to have a positive effect on students. When we got to the traditional rules and procedures, I heard students talking about how it “matched up” with what they were doing outside. Play with it.

Posted in basic math operations, fractions, math instruction, mathematics, remedial/developmental math | Tagged: , , , | Leave a Comment »

Metric to Metric Conversion: Ultimately, it’s a Proportion!

Posted by mark schwartz on July 27, 2016


This discussion came about because one student in one class simply asked “Why does this work”? He was referring to the procedure for converting metric units to other metric units, for example, “how many centimeters are there in 10 kilometers?” He could see that the “moving the decimal point” procedure worked but he kept insisting that there must be more to it; that somehow someone had figured this out and he wanted to know how it had been figured out. I had no clear answer to this and told him (and the class) that I would think about it. What I came up with isn’t necessarily the reality of the derivation of that procedure, but it did start with something that we had already discussed in class – proportions – and he was willing to accept this as a demonstration of why it works but wasn’t about to consider it a true explanation. Loved this guy!

The Story

In Colaw and Ellwood’s 1900 School Arithmetic: Advanced Book (page 252) is a discussion of the metric system. Among other interesting things, they note that kilo hecta and deka are Greek, while deci, centi, milli are Latin. In reading through their discussion, I got to thinking about how they, as we do today, convert one metric unit to another: a 7-point scale and simply “move the decimal point” … but some of their commentary made me think about how and why this 7-point scale works.

If asked how many decimeters are in .04 kilometers, one has a variety of strategies to use. If the 7-point scale (kilo hecta, deka, unit, deci, centi, milli) is known, one can write .04 at the kilo point on the scale and then visualize moving from kilo to deci, which would give a move of 4 places to the right. If the decimal point is moved four places to the right, this shows that .04 kilos is 400 decis. Typically, students are accepting of this ‘shortcut’ because it is much more manageable than other systems. But, the question was “why does it work?”

I believe the underpinning for the move-the-decimal method is to do the problem by first converting all the units to the amount at each point on the scale that equals one unit. This by no means is a rigid mathematical derivation but rather a way of demonstrating the relationships using a previously studied math relationship, namely proportions.

The traditional 7-point scale looks like this:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1000             100                  10                    1                      1/10                 1/100               1/1000

This scale shows the number of units in a named place-value. “Kilo” means 1000 units; “deci” means 1/10 of a unit, etc.. But let’s ask the question from the point of view of the unit: how many kilos would it take to make a unit? How many decis would it take to make a unit, etc.?

Here’s how the “unitized” 7-point scale would look:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1/1000          1/100                 1/10                 1                      10                    100                  1000

It appears as though the scale has been reversed, and it has because we are viewing the information from the perspective of what it takes to make one unit. For example, it can be read as “1/1000 of a kilo equals 1 unit” or “10 decis equals 1 unit”, etc. The point of this is that all of the place-value names are now on the same scale and having them on the same scale permits one to establish proportions.

For example, on this scale 1/1000 of a kilo equals 10 decis because they both equal 1 unit. Another way of stating this relationship is to state that “1/1000 kilos is to 10 decis”, which is a phrase describing the first rate of two rates that would make up a proportion. What would be the second rate? The original problem was “how many decimeters are in .04 kilometers?”

In this case, being consistent with the idea in proportions that the numerators are all the same type of units and the denominators are all the same type of units, what is seen is the relationship of kilos to decis, is:

kilo  1/1000  =  .04  = 400 decis
deci    10        x

It is this proportional relationship which provides the basis for conversions from one metric unit to another, as long as the units used are those that “equate” them to 1. Students must be comfortable knowing how many ‘dekas’ it takes to make one unit (since a ‘deka’ is 10 units, it takes 1/10 of a ‘deka’ to equal a unit). This may seem counterintuitive since we typically say, for example, that a kilo is a thousand units, which is true but the focus here is with how many kilos it takes to make a unit.

Given this discussion of the two methods, it seems most likely that students would tend toward the ‘move the decimal point’ system. It doesn’t require any computation. But the point of presenting both of these it to bring out the reality that the ‘easier’ system is based on a proportional system. Just another example of the power of proportions based on an interested student’s insightful inquiry.

Posted in algebra, basic math operations, math instruction, mathematics, proportion, remedial/developmental math | Tagged: , , , | Leave a Comment »

Rephrase That Impossible Application Problem

Posted by mark schwartz on July 19, 2016


As I was presenting a topic one day, a student said that what I was saying didn’t make any sense and could I please say it differently. My first reaction was to ask the class if it was true for them too; some agreed. It wasn’t a rude statement and I took the comment seriously and did rephrase what I said and asked if that made more sense and apparently I got it right. Then I got to thinking not only about that moment but other moments where what I was saying may not have made sense, but nobody bothered to stop me. As far as I can tell, it wasn’t the math content but the language I used to describe the content that bothered them. Thus the story that follows.

The Story

Question: Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?

Before considering the answer, consider if that question is the same as: how many years have I attended the city fireworks display on the fourth of July?

If your answer to the first question is 11, you’re wrong and as vague as the second question is, the answer is 11.

In both cases, which seems to be the same case, I suspect you got your answer by simple subtraction, 2016 – 2005 = 11. The thing to consider though is what exactly is being subtracted? Let’s bring this into a more manageable range, like 5 – 1. If you were to do this operation on a number line, you could put your finger on the five and move to the one, counting as you go and thus you would get 4. That four represents the number of movements from point 5 to point 1 on the number line. When you move from 5 to 4, you say “1”, in essence, “scaling” the distance between 5 and 4 as 1 unit, regardless of the actual distance. Given that it’s a number line, the distance between the points on the number line will all be the same. So, when we say 5 – 1, we are asking how many distances are there between 5 and 1. By the way, this distance analogy is similar to the idea of having 5 kittens and giving 1 away – how many kittens have you? In this case, it’s not distance, it’s kittens but conceptually it’s the same. We need not bother with scaling the number of kittens, because it’s a quantity not a distance, although some consider distance a quantity. As far as “how many years have I attended the fireworks?”, what’s being counted here is the number of years – an amount of time scaled rather than a distance. So, 2016 – 2015 is one, etc. as far as counting.

What’s the point? Remember the first question? To repeat: “Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?”

What is being counted here? Again, consider the number line. We’re not counting the distance between points on the number line rather were counting the number of points. The first question then has to be a subtraction plus one, which really is asking for the inclusive count.

You might say “so what?” to the difference between the first and second questions but looking at them as I did points out that there is a difference. The real issue here is the nature of asking questions in a math class. If we, as instructors, ask ambiguous questions, or questions which require students to reflect on the context of the information as well as the information in the question (and students don’t see the need to reflect on these issues) then we are, in a sense, misleading them and adding to their confusion about math. The context in this case is the words we instructors use.

I’ve seen this in questions in texts. We glibly accept the questions and answers at the end of the chapter and if some of those questions are questionable, we simply don’t assign them. But it’s not just the questions in texts. It’s how we state information, it’s how we use the language to structure questions and present concepts. The difference between the first and second question demonstrates this.

We should be attempting to be better at some precision in our questions and presentations because, like it or not, instructors are math role models for students. If we expect precision and accuracy from students, we should also expect that they can phrase good questions and it’s the instructor that establishes the idea of a well-phrased statement. It also seems it’s a critical component of being able to arrive at a correct answer to a problem. The caution to read “word” problems until you understand it is reasonable, but what if you never “understand” the problem? My thought is that students have to have license to and practice in rephrasing problems, without changing any of the relationships in the problem.

For example, when teaching percent using the percent proportion model (you can see how this is presented in the Percent Proportion posting in this blog), I point out to students that most percent problems can be rephrased. An example: A farmer sold 180 sheep, which represented 16% of all the sheep he had. How many sheep had he after the sale?

There are a lot of extra words in this problem, but only two numbers. I asked the class to rephrase this problem focusing on the relationship between the numbers. The students wrote all their attempts on the board, so that we could discuss the thinking that brought about their answer. What they produced, based on the percent proportion model that I presented to them, were two rephrased problem: First, “180 is 16% of what number of sheep?” to get the total number of sheep and then subtract the 180 he sold. The second was to see that if 180 sheep were subtracted from the total (written as x – 180), this would represent the number of sheep left and the percent left would be 84 (100 – 16). The rephrasing then would be “The number of sheep remaining is 84% of what number of sheep?” The result of this rephrased problem still is the total but again, simply subtract 180 sheep. If you want to play with it, the answer is 945. Even if the percent proportion model isn’t used, this rephrased problem is much more manageable.

But here’s how to set up both results using the percent proportion model.

180  16
 x   100      … solving, x = 1125 … total after sale = 945
x - 180  84
  x      100   … solving, x = 1125 … total after sale = 945

We practiced this rephrasing idea some more and I reminded them that they don’t have to rephrase every problem, but if the problem seems “impossible”, rephrase it.

Posted in algebra, Historical Math, math instruction, mathematics, proportion, Proportions, remedial/developmental math | Tagged: , , , | Leave a Comment »

What? That Much Percent Increase?

Posted by mark schwartz on July 8, 2016


I like coincidences. Particularly when they provide learning opportunities for my students. We had just spent time learning about percent and percent increase and decrease. The problems in the text were good but not really challenging. The coincidence was that I was reading John McPhee’s The Curve of Binding Energy (1974) and I’ll start the story with what he said on page 18.

The Story

“Thousands of miles of tubes, pipes, and other conduits were needed to create a network of flow wherein the gas could now go through a membrane, now return to try again, now go on to a new membrane, gradually advancing, in a process of separation and elimination, until what had begun as seven-tenths of one percent U-235 was more than ninety percent U-235 – fully enriched, weapons-grade uranium.”

I’d never heard this detailed an explanation of how weapons-grade uranium was made. But what really got my attention was that his statement could be a percent increase problem. I worked it out before I gave it to the class, rounding off the initial “more than ninety percent” to a manageable 90%.

Further, I decided that it would be an in-class extra credit exercise and allowed that the students had to first work within their assigned group, but once they had an answer they could discuss it with other groups.

I did this because the percent increase is 12,757% and this size percent increase would cause the groups to question what they did, even if they got that number. There were occasional answers to the problems in the text that resulted in percent increases of more than 100% but nothing quite like this. Once I gave them the problem and answered any preliminary questions and they got to work, I roamed the room listening to the strategies they came up with to do the problem.

The first issue was how to numerically express seven-tenths of one percent. One group asked if they could talk to another group to get help expressing it. So, I stopped the class and said that if they are willing to accept the following condition, they can work as a class to get the answer. The condition was that everyone in the class would get the same grade. They accepted. There was an eruption of conversation and as I roamed around, I was asked if what they got was right. I just referred them to other members of the class.

Once there was consensus on how to represent the initial percent, they simply continued with what they had learned about setting up percent increase problems. By the way, I taught a somewhat non-traditional method that doesn’t use a formula, rather it uses a somewhat modified percent proportion approach. You can look at it in this blog at Percent Proportion.

Several groups quietly called me over to show me their result, asking if they were right. Some were and some weren’t but I wouldn’t say yes or no, reminding them of the condition under which they were working. So, more talk, discussion and exchange of how to set up the problem.

It was interesting to watch the evolution of the shared work – people got up and moved around the room; some asked to and did use the white board; I heard a lot of “show me” and “why did you do that?” and “that doesn’t seem right”. But, ultimately there was class consensus on the right answer.

They did, however, insist that I walk through how I thought about it even though they got it. So, I put on what I call my “slow-motion-math” hat and gave them the following:

Ninety percent is 90/100, so the amount of increase is 90 – .7, or 89.3%. Seven-tenth of one percent is (7/10)(1/100) or 7/1000 (I did this because I saw a lot of questioning on how to express it). This in percent is .7/100 (or if you were sure, you could have just written .7 over 100). So the question can be put in a percent increase frame. First, the amount of increase is 89.3 and since it started at .7, the amount of increase relative to the beginning point can be expressed as 89.3/.7, or 893/7 (they questioned if doing this would give the same answer and we discussed this). Using the proportion statement 893/7 = x/100 gives a percent increase of 12,757%, rounded. So, doing the original problem led to some other related talk about fractions, decimals and rounding. Neat.

After all was said and done, I got questioned about this exercise because there was a sense that it was a trick question. I have noticed that when students feel uncertain about a math problem, the frequently asked question is just that. I then heard stories from the class about their prior math experiences where trick questions unfortunately were used to presumably teach them something about math, but the only learning was frustration because a lot of the tricks were beyond the bounds of what had been taught and in essence they quit. Given what I heard, I may have quit too. Somehow they concluded that math is just knowing the right tricks.

But once they were accepting that it was an interesting problem, I noted to them that as they read books, magazines, watch TV or come across “mathy” stuff, they might play with it as we did with this problem. And of course they noted to me to record the “A” for all of them.

Posted in basic math operations, Historical Math, math instruction, mathematics, percent, proportion, Proportions, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Two Alternatives to “Borrowing” When Doing Subtraction

Posted by mark schwartz on July 4, 2016


To do traditional subtraction one has to know the operation of “borrowing”. Most students can do it, but if there are double zeros (or multiple zeroes in either or both numbers), students find this troublesome. Further, most students don’t know the basis of borrowing. There is no sense of place value and also there is no awareness of what is being borrowed and what bundling and unbundling means. They simply follow (as best they can) the steps they were taught.

That’s one of the principle reasons why students are bothered by subtraction. And, sometimes, even after discussing place value and bundling and unbundling, there is still no significant change in students being able to do subtraction. Borrowing is bothersome.

But there are two alternative methods which don’t involve borrowing. Both of them involve an interesting operation and, in my view, this simplifies subtraction. The first alternative has been presented previously in the blog article Subtract by Adding but I wanted to present both of these ideas in the same article because, in essence, they are the same!

The Story

The first alternative: the 9s-complement

This is based on computer math. In the problem 203004 ̶ 044726, the subtrahend (the number being subtracted) will first be replaced with its 9s-complement. The 9s-complement is found by subtracting each digit in the subtrahend from nine. So 044726 becomes 955273. The leading zero in the subtrahend is included to assure that for that place value, it’s really 9 – 0 or 9.

The next step is to add one to 203004, making it 203005. This ‘one’ is the leading one that shows up in the answer. It is described as ‘dropping’ the leading 1’, but the reality is that it is added to the unit column in the minuend as the first step of the procedure. If this seems strange, I’ll demonstrate later why this is done.

 Now ADD these two numbers:  203005
                           + 955273 

In this case, drop the leading ‘1’, and you have the answer (“leading” means sticking out beyond the place values of the numbers in the problem). The mystery about dropping the one and the mystery of adding one to the minuend can be explained by demonstrating why this operation works. A problem with fewer digits will make it easier to follow the explanation. Use 312 – 67.

First add ‘zero’ to this problem in the following way: 1000 – 1000 + 312 – 67. Adding 1000 and subtracting 1000 doesn’t change the value of the problem. Replace 1000 with 999 + 1 and use the commutative property: 312 + 1 + 999 – 67 – 1000.

Continuing with the associative property and doing all the indicated operations:

(312 + 1) + (999 – 067) – 1000 = 313 + 932 – 1000 = 1245 – 1000 = 245.

This demonstrates (1) why 1 is added to 312, (2) why the 9s-complement of 067 is taken, and (3) why the leading ‘1’ is dropped.

The second alternative: the 10s-complement

The second alternative is very similar and uses the same complements method, but in this method the complement of 10 (not 9) is taken. This is based on operations on the abacus, not the computer complements.

So, back to the original problem, subtracting each digit in 44726 from 10 gives 66384. Then ADDING this 10s-complement gives:

          + 66384

In this procedure, there is no leading zero to include in the subtrahend and there is no 1 added to the minuend. In essence, the reason why it’s 66384 and not 966384 is because 10 ̶ 0 leaves a zero in that place value position.

We this isn’t the answer of 158278 which we got above. However, subtracting 1 1 1 1 1 0 from 269388 gives 158278 – the same answer as above. Again, this looks like magic but a demonstrate will again show why this operation works.

Applying this method to the simpler problem of 312 – 67, gives

       + 43

And subtracting 110 gives the answer of 245.

Or, as a student suggested, as each place value addition is done, subtract one from the answer, except in the unit column.

Why does this work? This is similar to demonstrating how the 9-complements works.

312 – 67

312 – 67 +110 – 110      (basically, add a zero in the form of +110 – 110.)

312 +110 – 67 – 110     (commutative property)

312 + (110 – 67) – 110   (associative property)

312 + 43 – 110

355 – 110

24 5


In different classes, we had interesting discussions comparing the two methods. The consensus was that the 10s-complement was easier, although it had two steps. It was easier because you didn’t have to remember to add one to the minuend or remember the leading zero in the subtrahend. But then I reminded them of what one student said about the 10s-complement, which supported their consensus.

Remember what the student suggested? He said “as each place value addition is done, subtract one from the answer, except in the unit column.” He realized that in the 9s-complement where each subtrahend value is subtracted from 9, this is exactly the same as in the 10s-complement when after subtracting each subtrahend value from 10, just subtract 1 more! Algebraically it’s (10  ̶  n)  ̶  1, giving 9 ̶ n. This student really understood both procedures!

Posted in basic math operations, math instruction, mathematics, remedial/developmental math, subtraction | Tagged: , , , | Leave a Comment »

Percent Problems from 1868

Posted by mark schwartz on June 28, 2016


After presenting percent and having students work many problems from the text, I decided to give them a set of problems from 1868. I did this for several reasons: first, I gave this as an in-class quiz in which I allowed the entire class to discuss strategies, compare answers, and work with others. I roamed the room to watch things happen. Second, I gave these problems because they all had fractions in them as well as related information that they had to decide where it fit in the problem. Problems like these are rare in today’s texts. It was a real challenge for the class but some of the students actually said it was fun! The story begins with a copy of what I handed to them. I’d like to note that I taught a visual percent proportion method only (no formulae, no short cuts) – Percent Proportion – which you can see in this blog. It’s a slower procedure but gives students a better understanding of percent. They said so.

The Story

Here’s the assignment I gave the class …

The 5 problems below are from The Progressive Practical Arithmetic by Daniel Fish. (Ivison, Phinney, Blakeman & Co, Chicago, 1868, pg. 228). There are some minor punctuation and word changes.

Please show all your setups and solutions. Consider carefully how to handle the fractions in the problems. Consider that only non-repeating decimals will give you a precise and accurate answer, so you may have to use improper fractions in your calculations, or use the “fraction” key if your calculator has one. Feel free to work with anyone (or everyone) in the class. Be careful handling dollars, cents and decimal points, and fractional answers. ALL ANSWERS ARE MONEY – DOLLARS AND CENTS.

Before you start, any questions?

  1. A miller bought 500 bushels of wheat at $1.15 a bushel, and he sold the flour at % advance (profit) on the cost of the wheat. What was his gain?

2. A grocer bought 3 barrels of sugar, each containing 230 pounds, at cents a pound, and sold it at percent profit. What was his whole gain?

3. A sloop, freighted with 3840 bushels of corn, encountered a storm, when it was found necessary to throw percent of her cargo overboard. What was the loss, at cents a bushel?

4. A gentleman bought a store and contents for $4720. He sold the same for percent less that he gave, and then lost 15 percent of the selling price in bad debts. What was his entire loss?

5. A man commenced business with $3000 capital. The first year he gained percent which he added to his capital. The second year he gained 30 percent on the whole sum, which gain he also put into his business. The third year he lost percent of his entire capital. How much did he make in the 3 years?


Roaming the room and hearing the discussions and debates was a lot of fun. I occasionally answered questions with questions on how to handle the values and the relationships. As it turned out, when the work was turned in, everybody got it! And, of course, the class wanted this format for all future quizzes, but I noted to them that I had other formats in mind. Someone pointed out how valuable this learning experience was and why would I not want to repeat it? Ah, logic, but as it turned out the other quiz formats turned out to be good learning experiences too.

Below are the answers if you want to play with these 5 problems.

  1.    $95.83 &1/3 (I accepted $95.83; our money system stops with pennies)
  2.    $10.35
  3.    $900
  4.    $1209.50
  5.    $981.25

Posted in fractions, Historical Math, math instruction, mathematics, percent, proportion, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Fibonacci: Surprise and Pattern in Mathematics

Posted by mark schwartz on June 18, 2016


Teaching math involves a lot of things, first of which is “do I have their attention?” and secondly, “how do I keep their attention?” It took me a while to realize that these were the wrong questions. The real questions are “does math have their attention and does it keep their attention?” There is a difference. We’ve all developed presentation modes, perhaps several of them, tailored to the content and the class … we do things that feel comfortable but also things that seem to keep students’ attention. Let me show you one that I’ve used in a developmental class that gets conversation going beyond the actual exercise and leads to two things important to math that I like to emphasize as the semester continues: surprise and pattern.

The Story.

An interesting way to demonstrate surprise and pattern is the use the Fibonacci series. Don’t teach it, just use it. In fact, in some instances it’s never mentioned but in most instances, students want to know how the exercise came out the way it did.

The first thing is to have students work in groups of two or three. This group behavior engages each student in the process of deciding who is going to do what part of the activity, and as they work together they realize that working together is actually a fun math experience. I typically randomly assign students to groups in an effort to get them to work with someone new and this involves their moving around to sit with their new partner. This activity alone is typically something that they’ve never experienced in a math class. I give them time to introduce themselves to each other.

Once all the hubbub of moving and introductions is over, I ask them to get out their calculators or phones that have calculators because they’re going to need them for this exercise.

I then ask them to designate a writer and a “calculator” and ask that the writer get out a piece of paper and make a list down the page from 1 to 25. When everyone is ready, I then ask each group, one group at a time, to give me two numbers from 1 to 9, inclusive. Sometimes the word “inclusive” needs a little explanation. The writer is then to put their pair of numbers as the first and second number in the list. If the question is asked about the order of the numbers, tell them that it’s their choice. I then write their choice on the board. I do this because I want each group to have its’ unique set of two numbers and also if they see what’s been selected it will help avoid duplicates. Once every group has done it, I ask them if they’re ready to go on.

The instruction is to add the first and second numbers to get the third number. Then take the 2nd and 3rd numbers to get the fourth and continue this way until they’ve filled in their list to 25 numbers. I take one of the pairs from the board and show them how it works. I again ask if there are any questions before they continue. I point out that as they move down the list the numbers may get big and the job of the calculator is to get the numbers correct, so work slowly. As an aside, this sometimes leads to the group members deciding that they will all do the calculations to check on each other and make sure they’re right. If no more questions, get to work. I roam the room and watch them work and comment occasionally.

After all the groups have their list of 25 numbers, I ask them to slowly and carefully do these things: divide the 25th number by the 24th number, ignore the decimal point, and write down only the first 4 digits from the left (sometimes I have to show them what this means).

After they’re all done, I announce that I will now tell each group what their 4 numbers are. Using the list of their pairs on the board, I start with announcing 1618 for the first two groups and then I quit and announce that they all have the same number.

This is the first surprise. They typically verify with each other that this is true. Usually someone will ask how that happened and this begins the conversation; how can each group start with two different random selected numbers and yet come out with the same number? Enjoy the conversation and let them roam around for a while; they might hit on what happened. If they ask you for the answer, don’t give it but you can provide a hint that the answer is hidden in the process; can they see any patterns?

This first day exercise lays the foundation for students to realize that a lot of the math they’ll be doing will have surprise and pattern and that as we go through the semester, I’ll be referring to this a lot (which I do; after all, rules, formulae and algorithms are “frozen” patterns, aren’t they)?

And finally, when the class is over and they’re gathering their stuff to leave, there are animated conversations about what they just did and that’s very satisfactory to me. You might want to experience it too.

Posted in basic math operations, Fibonacci, math instruction, mathematics, remedial/developmental math | 2 Comments »

Some Old Commentary on Today’s Common Core Math

Posted by mark schwartz on June 11, 2016


The quotes herein are extracted from prefaces and introductions of texts ranging in date from 1839 to 1911. I’ve done this because then as well as now there‘s concern that math instruction and learning can always be made better, that students can perform better. Then, as opposed to now, they didn’t have nation-wide and international standardized tests and data on which to make their assessment of the need for change. Rather, they talked to each other and were motivated to improve instruction based on how students were performing. Even then, hints of the rationale behind common core math existed and it can be seen in the language. I’ve underlined select words and phrases that seem like those used to support common core today, but don’t limit your reading to what I underlined … you may have underlined different things. Italics are theirs.

If you would like more information about any of the authors or any of the books, write to me at . Reference the author, date and ”#n” (after the author and date).

The Story.

The design has been, to present these in a brief, clear and scientific manner, so that the pupil should not be taught merely to perform a certain routine of exercises mechanically, but to understand the why and wherefore of every step. Joseph Ray, 1866. (#1)

It is about 20 years since the first publication of the Elementary Algebra. Within that time, great changes have taken place in the schools of the country. The systems of mathematical instruction have been developed, and these require corresponding modifications in the text-books. Charles Davies, 1874. (#5)

Explanations rather embarrass than aid the learner, because he is apt to trust too much to them, and neglect to employ his own powers; and because the explanation is frequently not made in the way, that would naturally suggest itself to him, if he were left to examine the subject by himself. The best mode, therefore, seems to be to give examples so simple as to require little or no explanation, and let the learner reason for himself, taking care to make them more difficult as he proceeds. This method, besides giving the learner confidence, by making him rely on his own powers, is much more interesting to him, because he seems to himself to be constantly making new discoveries… this mode has also the advantage of exercising the learner in reasoning, instead of making him a listener. Warren Colburn, 1839. (#7)

The aim of this treatise is to meet, more fully than has been done heretofore, the requirements of the highest standard of mathematical instruction in the best high schools and seminaries. To this end, great care has been taken to include all the more important parts of analysis, to treat each topic with as much conciseness as is consistent with clearness and elegance, to introduce valuable original processes, and to secure throughout an arrangement most conducive to a philosophical development of the science. Benjamin Greenleaf, 1864. (#10)

That useful mental discipline may be attained, the theory and principles of numbers have been clearly presented, and problems have been given requiring thought and discrimination. The inductive plan has been followed throughout, principles have been developed from methods, rules derived from analyses, and oral and written exercises combined in a rational manner.  Benjamin Greenleaf, 1881. (#12)

The transition from the traditional algebra of many of our secondary schools to the reconstructed algebra of the best American colleges is more abrupt than is necessary or creditable. This lack of articulation between the work of the schools and the colleges emphasizes the need of a fuller and more thorough course in elementary algebra than is furnished by the text-books now most commonly used. Charles Smith, 1911. (#13)

The inductive method is applied throughout the book. New topics are introduced by carefully prepared questions and suggestions designed to develop a correct understanding of the principles to be taught, and to give a clear insight into arithmetical processes and relations.  John Colaw and J. K. Ellwood, 1900. (#16)

Here the object is, first, to lead the learner to see clearly the distinction between the process of reasoning he is led to pursue in the solution of a question, and the numerical operations he I required to perform as the result of that process. In order to do this, he is required to perform numerous questions, retaining the operations as he proceeds, and leaving them all to be performed at last, when the reasoning process has reached its conclusion. In this way he is led to see that the reasoning process is precisely the same, and the operations to be performed precisely the same, for all questions which differ only in the particular number that are given, that thus, in fact, he has obtained a general solution of his question. William Smyth, 1859. (#19)

When the more systematic treatment of the science is presented, the pupil is led by natural, progressive, and logical steps to an understanding of the definitions, principles, processes, and rules, before he is required to state them; consequently, all definitions, principles, and rules are but the expressions of what he already knows. It is evident, therefore, that the plan pursued in the work will develop in the student the habit of investigating for himself any subject which may claim his attention, and this is an extremely important part of proper teaching. William Milne, 1893. (#20)

There are two general methods of presenting the elements of arithmetical science, the Synthetic and the Analytic … Analysis first generalizes a subject and then develops the particulars of which it consists; Synthesis first presents particulars, from which, by easy and progressive steps, the pupil is let to a general and comprehensive view of the subject … Synthesis constructs general principles from particular cases. Analysis appeals more to the reason, and cultivates the desire to search for first principles, and to understand the reason for every process rather than to know the rule. Horatio Robinson, 1863. (#23)

The science of Arithmetic, until somewhat recently, was much less useful as an educational agency than it should have been. Consisting mainly of rules and methods of operation, without presenting the reasons for them, it failed to give that high degree of mental discipline which, when properly taught, it is so well calculated to afford. But a great change has been wrought in this respect; a new era has dawned upon the science of numbers; a “royal road” to mathematics has been discovered, so graded and strewn with the flowers of reason and philosophy that the youthful learner can follow it with interest and pleasure …   Edward Brooks, 1873. (#27)

It is the purpose of the book to lead the young to comprehend and appreciate mathematical reasoning, as well as to solve problems. Edward Olney, 1874. (#30)

Very many people will prefer to have the student trained to be rapid and accurate in computations, and they will esteem a rapid accountant more competent in mathematics than the learned astronomers of our time; while others will prefer that training which cultivates the reasoning powers, even at the expense of practical expertness in the use of numbers. William Milne, 1892. (#31)

The plan of this work is, first, to give a course of reasoning leading to those conclusions from which the rules are drawn – and this is given in language free from perplexing technicalities, and easily to be understood. Secondly, to give in plain and comprehensive language, the rule drawn from such reasoning. Thirdly, to give examples for practice in application of the rule given. Fourthly, to introduce, at proper intervals, miscellaneous questions, involving the several rules which shall have been passed through. The explanations are so written as to throw the student into the place of the original reasoner, as plodding his way through, until he arrives at a conclusion from which he can draw the rule for himself. Cornell Morey, 1856. (#38)

The explanations of the written processes are not designed to serve as models for the pupil to memorize and repeat. They are intended to supplement the analysis. In some cases, a formal analysis is given; in others, a principle is deduced or demonstrated; and in others, the process is described or its principals stated. Neither teacher nor pupil is denied the privilege of determining his own explanations. E. E. White, 1870. (#45)

Posted in Historical Math, math instruction, mathematics, remedial/developmental math | Leave a Comment »

Math Fragments Perpetuate Fragmented Learning

Posted by mark schwartz on June 8, 2016


Carla (not her real name) indicated that she had missed a lot of education because in the religious school she attended, time was spent on theological not academic issues. She has very fragmented math information and it seems that she tries to take any new material and fit it into the fragments she knows. She seems to have a need to preserve what she knows, not realizing that it interferes with learning. It is likely not a conscious act.

The Story.

An example of this is in her work on a quiz. I gave a bonus question. Typically, they aren’t too hard but do incorporate some material that has been learned but also include something that if you are aware, makes the problem somewhat easy.

The problem was

.25 ─ 1/2  + .4 ─ 1/4  + .5 ─  2/5.

Her work and her answer was

51/20 + 17/20 + 27/20 = 95/20 = 19/20

These numbers were puzzling until I realized that she had seen the problem as

25 1/2 + 4 1/4 + 5 2/5

The first thing to notice is the blurring of the “—“ sign. It disappears, thus giving her 3 mixed numbers to add.  She said that she saw the minus sign as dashes separating the numbers from the fraction.

Somehow one of the fragments she recalled was that mixed numbers can be converted into improper fractions, so the weight of that fragment also drove the disappearance of the “─”.  It is also possible, since it had been noted in class, that the notation of a mixed number “A b/c” really means “A + b/c”. This fragment seems to have stayed, but the “+” got replaced with “─”.

The next fragment that emerges is that in order to add fractions, you have to have a common denominator. I think this explains why there is a “20” in each denominator, rather than the 2, 4, and 5.  There was an awareness of this and she found the correct lowest common denominator, but the fractions weren’t converted to 20 correctly.

The decimal points seem to have been totally ignored; when we talked about this, she said she had never seen them before and when she saw them, she decided it was a printing error.

Finally, there is a fragment about reducing fractions. It was demonstrated that “canceling” really is looking for a common factor in numerator and denominator. Somehow, only the part about the numerator was operating.

Carla really wanted to learn math; she was motivated and trying very hard and I applauded her effort but talked to her about fragments and how the fragments were dominating her learning. When we talked about this she kept clinging to the fragments she knew because she had seen this material before and was sure that what was going on now had to fit into what she already knew. After presenting her with visual and kinesthetic procedures which didn’t offer her the chance to use her fragments, she slowly but surely started to restructure what she knew, rather than trying to restructure the new information to fit her fragments.

Recognizing a student’s learning problem sometimes has nothing to do with what we see, but rather is an invisible impediment to learning … and not easy to find, so keep snooping around.

Posted in basic math operations, fractions, math instruction, mathematics, remedial/developmental math | Leave a Comment »