Tag Archives: word problems

Caulfield on the End of Word Problems

Frazz, my current favorite comic strip, had a lovely series this week on the end of word problems. My favorite:


The end of word problems redux

Dan Meyer linked to and quoted from my screed on the end of word problems the other day. This led to some robust discussion on his blog, which I now take the opportunity to reply to.

jg writes:

the lion’s share of the difficulties isn’t the silliness of the problems, no matter how silly they are – it’s the illogical, poorly defined, and trite mental worlds that most folks live in! _That’s_ our challenge!

One of the most useful pieces of educational research comes from the folks at the University of Wisconsin. Cognitively Guided Instruction (CGI, about which I have written a couple of times) demonstrated quite definitively that young children think quite a bit about mathematics-especially early number concepts, and addition and subtraction. What they don’t do is think like adults. This they need to be taught. But it’s not “procedure first then word problems”. It’s “procedure is an abstraction that follows from students’ informal ideas that come from their interaction with the every day world”.

waldo writes:

I tell the kids pure math is strength training for the brain. Just because you never do bicep curls while in a “real sports” situation, it doesn’t mean that doing those curls is irrelevant to your sporting ability.

This is such a tempting analogy but we really don’t have a shred of evidence that it’s so. What I learned from reading Thorndike, the godfather of transfer, is that practice on certain skills makes one better at those certain skills and on nothing else. We have never been able to demonstrate that learning thing A makes you better at thing B. Even so-called “near transfer” is unsubstantiated. What waldo points to is “generalized transfer”-a harder nut to crack and completely unproven to exist.

Now there are lots of other reasons to study pure mathematics. And I have no objection to the idea at all. Mathematics, like poetry, is a beautiful achievement of human intellect for instance. But the cans of peaches problem is not one from pure mathematics. And it’s not from applied mathematics either. It is a word problem. A clever little puzzle, to be sure, but not a useful representation of the subject.

Mark Schwartzkopf writes:

I’m not sure what could be meant by “thinking mathematically”. Translating words into algebraic expressions, without the need to understand the situation fully IS thinking mathematically.

The emphasis was intended to be on thinking, not on mathematically. Recall the Johnson quote was, “students having difficulty can learn basic procedures even if they are unable to reason out a problem.” I was objecting to the author’s satisfaction with getting students to apply procedures without reasoning (thinking).

And I’m not so sure I agree that “translating words into algebraic expressions…IS thinking mathematically.” Especially when the words are written with this translation being the only purpose. If “of” always means “multiply”, am I really thinking mathematically when I rewrite “1/2 of 3/4” as “1/2 x 3/4”?

Schwartzkopf continues:

Before 1500 or so, the science of math was developing at a snails pace. It was extremely hard to think about. So much so that people would have to travel to other countries in order to learn the arcane skills of multiplication and division. At this point, math and algebra texts were pretty much exclusively word problems; algebraic notion had not been invented yet. As the mathematical community began to develop the means of translating word problems into algebraic notation, math became way easier, and began to develop at a faster and faster rate.

It is absolutely the case that algorithms and memorized procedures free up the human mind to tend to other, more important matters. I have no issue with algebraic symbolism, nor with its use in K-12 classrooms. My beef is with curriculum that offers students little of intellectual value and little in the way of honesty about the actual uses of the subject.

Cardinal and ordinal numbers

NOTE: I get a lot of hits from Internet searches on “cardinal and ordinal numbers”. If that’s how you got here, consider going directly to an expanded and illustrated version of this post on Sophia. It is a better version of the same ideas that are here.


In my math content course for future elementary and special ed teachers, we are working on the difference between cardinal and ordinal numbers. I promised them a reading on the topic so they could work on it outside of class. But the first three pages of Google results (and really, who goes deeper than that?) all return the same basic idea, which is not as nuanced as the one I need to communicate to my students. So here is my take on things. If you find it useful, link to it please so that future students (mine and others’) will find it more easily.

introduction to the topic

The basic introduction to ordinal and cardinal numbers is this: Ordinal numbers refer to the order of things. If I was fifth in a race, fifth is the ordinal number. Cardinal numbers refer to how many things there are. If there are five fingers on one hand, five is the cardinal number.

At the introductory level, the distinction can be made linguistically: first, second, third are ordinal numbers while one, two, three are cardinal numbers.

This is the distinction that is made on each and every one of the websites I looked at this afternoon. Here is a typical example.

A subtler view of the topic

What do we mean when we say that “the Minnesota Twins are number 1”? We mean that they are the best, or that if we put all baseball teams in order from best to worst, they would be first.

The claim that “the Twins are number 1” is a claim about ordinal numbers. That is, we don’t always make the linguistic distinction. Sometimes we say one when mean to refer to a cardinal number (the Twins are one team in the league), sometimes we say one when we mean to refer to an ordinal number (the Twins are number one in the league).

A problem

In class, we are reading a lovely book titled Children’s Mathematics that reports results from the Cognitively Guided Instruction (CGI) research project at University of Wisconsin, Madison. The premise of the project is that young children come to school with powerful mathematical ideas, but that these ideas may differ from an adult’s way of thinking. The better teachers understand their students’ ideas, the better basis the teachers have for making instructional decisions. CGI set out to document the ways students think about addition and subtraction problems, and to associate these ways of thinking with strategies students use to solve problems.

The book comes packaged with a CD-ROM of second-grade children solving problems of different types, and demonstrating various strategies. One major strategy is counting up/back.

Consider this problem:

Problem 1. Griffin had 10 apples. He gave 3 apples to his sister Tabitha. How many apples does Griffin have left?

Problem 1 standard solution. The standard counting technique is for a student to say, “He had ten apples. Nine, eight, seven. He has seven apples left.”

Problem 1 alternate solution. In one of the CGI videos, a student solves a similar problem in this way, “He had ten apples. Ten, nine, eight. Take away that; it’s seven.”

In the standard technique, we say “nine, eight, seven”. These are cardinal numbers. They represent how many apples are left after Griffin gave away each apple. He gave away one apple NINE, he gave away another EIGHT, he gave away another SEVEN. So after giving away three apples, there are seven left. Because we are counting back using cardinal numbers, the last number we say represents the number of apples remaining.

In the alternate solution, we say “ten, nine, eight”. I argue that these are ordinal numbers. They represent which apple Griffin is giving away at each step. He gave away apple number TEN, then he gave away apple number NINE, then he gave away apple number EIGHT. So after giving away three apples, he has given away apple number eight, there must be seven left. Because we are counting back using ordinal numbers, the last number we say is one bigger than the number of apples remaining.

A lovely question

A student asked a lovely question in class. Paraphrasing, she asked, “What if he were counting up? How would his strategy be different from the usual one?”

Surprisingly, his strategy wouldn’t be any different.

It would be highly unusual for a student to solve Problem 1 by counting up. So let’s consider another problem:

Problem 2. Tabitha had 3 apples. She picked some more. Now she has 10 apples. How many did she pick?

Problem 2 standard solution. The standard counting technique is for a student to say, “She had three apples. Four, five, six, seven, eight, nine, ten.” When the student says “four,” she puts up one finger then another for each successive number. After saying “ten,” she counts her fingers and says, “She picked seven apples.”

The numbers four, five, six, etc. are cardinal numbers. They each refer to how many apples Tabitha has after she picks another apple.

Or maybe they are ordinal numbers. Does “four” refer to Tabitha having four apples, or does it refer to the fact that the one she just picked is apple number four? Does four describe the set of apples, or does it describe the fourth apple? We can’t know from the information given. The ordinal and the cardinal counting sound identical when counting up.

And that is what makes the subtler distinction between ordinal and cardinal numbers so challenging. The easier distinction is based on the form of the words (fourth-ordinal; four-cardinal). But not all fours are cardinal.

When children are learning to count, they learn the sequence of words first: one, two, three. They do not fully understand that these words represent how many things there are. In that sense, it could be said that most children learn ordinal numbers first. The idea that the last number we say represents something about the whole group of things-that this is a cardinal number-comes later. It is not unusual to give a group of eight things to a 3-year old for her to count, to have her do so correctly, then to ask, “So how many are there?” and to have her say “twelve” or some other wrong answer. She has ordinal numbers-ascribing the correct number word to each object in the set; she does not have cardinal numbers-ascribing the correct number to the whole group.

How long until Twitter is exhausted?

I wondered this afternoon, and am now soliciting answers to, the following question: Given the 140 character limit on tweets and given the current (and presumable future) massive popularity of the service, how many years until every possible tweet has been tweeted?

Please state all assumptions relevant to your solution.

Inverse operations

A fellow math teacher and writer, Whit Ford, wrote recently about inverse operations. I found his approach interesting and refreshing because he was working on the mathematical relationship between addition and subtraction, and he had set things up in a way that suggested he wanted to look for parallels with other inverse pairs such as multiplication/division and exponentiation/logarithms. Among the principles he cited that I think he’ll do a nice job of generalizing are these:

[Inverse functions can:]

not share convenient properties with the “forward” version of the function, like being commutative or associative

– have ranges that force us to expand the universe of numbers that we had been habitually using before being introduced to the inverse function

– be more challenging to describe verbally than their “forward” sibling

This is lovely stuff which is too often missing in mathematics instruction, or that is at best tangential to instruction. He also suggested three conceptions of subtraction by discussing possible meanings for the expression 5-3:

[1] “five take away three”, or

[2] “what I must add to three to get to five”, or

[3] “five plus a negative three”

In a comment I suggested these additional conceptions:

[4] How much more is 5 than 3?

[5] How far is 5 from 3 on the number line? and

[6] 3 is part of 5; what is the rest?

I’ll expand on these here, and consider how all of this relates to the larger question of inverse functions and operations.

There has been a long-term research project at the University of Wisconsin, Madison called Cognitively Guided Instruction (CGI). That project was founded on the principle that if teachers better understand how their students think about mathematics, they will be more effective teachers. One of their early major results was documenting that children have a variety of strategies for solving addition and subtraction problems before they have been taught addition and subtraction. What is more, the types of situations children encounter in these problems influence what strategies they use more than whether it is formally an addition or a subtraction problem.

In particular, they identified four major categories of problems based on children’s problem-solving strategies: Join, Separate, Part-Part-Whole and Comparison. Briefly, a Join problem involves two sets being joined together; a Separate problem involves one set being separated into two sets; a Part-Part-Whole problem involves two parts making up a whole, but without any physical action joining them together; a Compare problem involves comparing the sizes of two different sets.

Depending on what is known and unknown in the problem, each of these four types encompasses problems that can be solved with subtraction or with addition. The CGI argument is that student strategies correlate closely with these problem types, rather than with the more formal categories of addition and subtraction.

So subtraction conception [1] above, from Whit’s original list, is a Separate problem. [2] is a Join problem. [3] is Join with negative numbers, which is beyond the scope of the original CGI work.

I wanted to add the other categories: [4] is a Compare problem. [5] is beyond the scope of CGI, which worked with set models (e.g. marbles instead of number lines) but is probably most like the Compare problems. [6] is Part-Part-Whole. Note that [1]-[3] all involve physical actions while [4]-[6] do not. Whether or not there is a physical action in a problem is very real conceptually for young children (and probably beyond!)

Now, from addition we get repeated addition, which is the main conceptual entry point to multiplication. Its inverse is division. Whit’s three generalizations about inverse functions-that they tend not to share convenient properties with their originators that they necessitate new numbers, and that they can be conceptually more challenging, are likely borne out in the multiplication/division example. It would be equally interesting to consider what the major categories of multiplication and division concepts should be. Depending on whom one asks, there can be as few as two categories for division or many, many more.

Next, from repeated multiplication we get exponentiation. Its inverse is logarithm. Once again, Whit’s generalizations apply. But I have not seen anyone try to categorize concepts of exponentiation nor logarithms. Perhaps this is because we have reached some threshold of abstraction in which people only think about these concepts in a formal way. But that seems unlikely-few people think only formally and most people think informally before they can think formally.

And this is the greatest lesson of CGI-if our informal conceptions of addition and subtraction influence how we solve problems at an early age, isn’t also likely that our informal conceptions of other operations influence how we solve problems later on?