Does your student know 10/9 is a fraction?

American elementary and middle school mathematics programs are poor in myriad ways. They lack breadth and depth. They lack reasoning. They lack precision. An object lesson in all of the above is the common use of analogy to teach math, even in grades 5-8.

 Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analogous connection anyway, even if they can abstract away from the literal -- something most children cannot do. And if you are are ELL, it is probably entirely worthless. Plus, the analogy is by definition imprecise, so its correctness will break down with even the slightest scrutiny.

 You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "a fraction is part of a whole".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:

•  1. "there's no such thing as ten ninths." that's the majority response in classroom after classroom. Why? Because a fraction is PART of a whole. How can a part of a whole be bigger than the whole? What's the whole then?  
• 1b. therefore, they believe no fraction can be bigger than 1.
• 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6. 
• 3. "three thirds is A Whole." Not one. 
•  3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?" 

 Additionally they don't know decimals are fractions. how could 1.2 be a fraction? Twelve tenths isn't a fraction, remember?

 These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.

 The correct explanation is that a fraction is a number. What number? A number defined on the number line as follows:

1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.

4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.

 Yes, teachers will need to build up to this. They should do so.

more from Wu

from Allison:

Going forward from what Catherine posted, Wu argues that most of what teachers know about fractions, and teach to their students, is wrong in the sense that it is unsupported by what the students have previously been taught.

Wu makes clear that you must be ruthless with your self when teaching. You must not allow yourself to use any concept you've not explicitly given your students. And almost all textbooks violate this rule in almost every lesson on fractions.

Nearly all textbooks avoid defining a fraction. Since they aren't defined, they don't define how you operate on fractions in terms of any real definition. Beyond that, when they do introduce operations on fractions, the books don't define operations in terms of what students know either. They teach rules without ever teaching where the rules come from. Their "explanations" invoke properties no one taught the students, but are just asserted. That means logically, their explanations are wrong, as you can't prove a statement by assuming the statement true.

For example, how does one justify multiplication of fractions? (I'll answer how later, but for now, discuss amongst yourselves.)

Yes, you can learn to use the rule properly without understanding where it is from, but you'll only go so far, and more, you will mistrust your teachers and mistrust math. Math will seem to be an endless series of magical statements with no relationship, and sooner or later, you'll run out of time/space/effort for keeping track of them. Properly understood, math is coherent, because every new step you take follows from the ones before.

Wu's fractions: Common Denominators

This is the 3rd in a series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I and Part II.

Last time, we left off after how to represent fractions on the number line. We showed that while a number is unique, and has a unique position on the number line, there were many different representations for that number based on how the number line was partitioned into pieces.

Recall we showed that 4/3 is the same as (5 x 4)/(5 x 3) with this example: locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:


Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20.



While our example was specific, our technique was not. Any fraction would have worked, and any new non zero partitioning. This demonstrated for us the mathematical fact: for all whole numbers k, m, and n, (such that n is not equal to zero and k is not equal to zero), m/n = km/kn.

In other words, m/n and km/kn are equivalent fractions. The equality symbol above tells us that these two quantities are the same. We can understand that to mean that there is one location on the number line that represents m/n, and it's the same location as that for km/kn, for any k, m, and n (such that n and k are nonzero.)

We note here that this is a good place to reinforce your comfort with commutativity of multiplication. We want students to feel comfortable recognizing that we could just as easily have said m/n = mk/nk and that would also be an equivalent fraction. This is where mastery of the underlying multiplication table is so important. The way to really believe the commutativity is to already know it's true for all of the natural numbers. We want to be able to multiply a fraction by k/k from either side without confusion.

This is also a good time to discuss a fairly beautiful fraction: k/k. Remember that our definition of a fraction is: the fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks. So the fraction k/k is the point on the number line when we partition each unit into equal kths and then make k jumps to the right of zero. This ALWAYS lands us at 1. So k/k is the same number as 1.

For students that are clicking this together, this is another way to see why m/n = mk/nk, but remember, we didn't resort to that explanation in the first place, because we'd like to become familiar with manipulating expressions like mk/nk WITHOUT reducing them. We are trying to build up more helpful denominators, to learn how to create new denominators, not just reduce them. Real mastery of k/k doesn't just lead us to say "that's 1, that cancels" but to say "we can replace 1 with k/k!"

Equivalent fractions are useful because they allow us to move quickly to compare fractions to each other, by use of this fact: Any two fractions may be represented as two fractions with the same denominator.

Mathematically, how would you do this? Well, take 2 fractions: m/n and k/l. m/n is the same as ml/nl. k/l is the same as nk/nl.
So now both m/n and k/l can be represented with the denominator nl.

Why do we care? Because now we can easily compare fractions, and we will now be better able to add and subtract them. Consider first the comparison. Two fractions with the same denominator can both be represented easily on the number line without confusion. Start with m/n and k/l. They become ml/nl and kn/nl respectively. ml/nl and kn/nl are represented by locations on the same sequence of (nl)ths. ml/nl is to the left of kn/nl if ml . By converting to a common denominator, we were able to solve the problem without having to graph out two different denominators. All we did was look at the numerators.

This leads to a specific method for comparing fractions. We convert each fraction to having the same denominator, and then look to see which numerator is bigger. In other words:

for all whole numbers k,l,m,n, k/l= m/n is equivalent to kn = ml .
This is just what we said above. It is also called the Cross Multiplication algorithm. (multiply the bottom of the left and the top of the right; multiply the bottom of the right and the top of the left. Which product is bigger?) In that case, we shorten the original procedure by ignoring that common denominator nl and just comparing the numerators. But indirectly, we are exploiting the fact that these two fractions are now represented by the same common denominator to determine which is larger or smaller.

Just as comparing two fractions is easier when you have common denominators, adding and subtracting them is easier too. We'll pick this up in the next post.

Update: above links fixed and should point correctly now!

More fun with number lines and fractions

Attempting to understand fractions as numbers, and that arithmetical operations on fractions follow naturally from arithmetical ops on whole numbers, we started to familiarize ourselves with arithmetic on the number line. Picking up from where I left off in this previous post, we remind ourselves that the critical key is not to be counting the Fence Posts, but the jumps. We could explain 7 - 4 by jumps, rather than counting the hash marks on the number line. Once that is crystal clear in our minds, we can then abstract away from the jumps, and think instead of motion on the number line as continuous motions, where we moved along the number line progressively:

This idea is a kind of intermediate step. For now, let's think of this motion to facilitate us thinking about the number of Unit Segments we've moved. That is, instead of thinking of addition in terms of jumps, we can think of addition as a continuous motion whose number of Unit Segments corresponds to the number. What's a Unit Segment? It's the length of the segment between 0 and 1.

We add 4 and 3 by starting at Zero and moving 4 units to the right. Then we move 3 units to the right. We get 7. Graphically, we see that on the number line, we reached 7 by moving right for 7 segments. So 7 is made up of 7 unit segments, just as it was reached by 7 jumps.


Subtraction, then, means moving the correct number of unit segments to the left. 7 - 3 means moving 7 units to the right, and then moving 3 units to the left. The number we reach, 4, is made up of 4 unit segments, just as it is 4 jumps from zero.


Another way to look at this is that the number 7 can be represented EITHER by the position of the point on the number line, OR by the length of the segment from 0 to 7. They are equivalent representations (isomorphisms, actually.) (It's not clear to me that this is the most child-friendly way to think about it. Someone with 4th graders needs to find out if it's useful to present this to children--I think it's easier to think of the jumps being of the unit length for now, and we'll adapt the lengths of the jumps.) Wu puts this as "we identify this standard representation of the number with its right endpoint" of the segment that starts at Zero. But if this is too hard, the jumps still work: the number is represented as the location on the number line that we end up at after we've completed all jumps.

We are now ready for fractions. No, really, we are! We begin with a specific fraction, and we work up to a generalization.

The fraction 1/3 has a numerator and a denominator. We will show how it is a fraction of our unit segment, the segment from 0 to 1. We now partition our unit segment into thirds. 1/3 is then symbolized by starting at 0 and moving to the right one partition of one segment:

The position of the right endpoint of that segment is the location of 1/3 on the number line.

But just as we could partition the unit segment into thirds, we can partition all of the segments between consecutive integers into thirds. The hash marks of these partitions are the fractions whose denominators equal 3. Again, this is saying "we identify this standard representation of the number with its right endpoint." These thirds function as our new "units", and we could make jumps on them just as before. The place we land on our jump is the number.

So, now that we know how to imagine the number line in thirds, we can generalize:

The fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks, or equivalently, partition each unit into n equal segments, and then move m nth-segments to the right.


There you have it--a definition of a fraction! Now, an important point glossed over in these examples is that a point on the number line is unique. Its representation, however, can be varied. This is worth stressing: a number, be it whole or fraction, corresponds to the unique location on the number line (or, as Wu would put it, to the unique line segment whose right endpoint ends at that location.)

With whole numbers, most people don't think that's very interesting. But with fractions, there are many representations for the same number. This can be confusing. The best way to clarify it is to say: they are still the same number-it's just a different way of reading the same number. This leads to a clear understanding of equivalent fractions.

Consider the fraction 4/3. we will show that (5 x 4)/(5 x 3) = 4/3 as follows: First, locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:


Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20. We could do this for any partition--7ths, 162nds, etc. It doesn't matter. No matter how you partition into equal bits, you haven't moved off of your point on the number line. Nothing about repartitioning changed your location.

Similarly, improper fractions are a breeze: we can see that 10/3 is 10 jumps to the right on the 3rd-hash-marks. It's also 3 jumps on the Unit Hash marks and 1 jump on the thirds: 3 and 1/3. Since the same point on the number line can be read in either fashion, they must be the same number, because a number is defined by its unique point on the number line.

We'll pick up with common denominators next.

On our way to fractions: the number line

Prof. Wu's document Critical Concepts for Understanding Fractions tries to clarify operations on fractions by use of the number line. Wu stresses that fractions should be defined succinctly as numbers (rather than as pieces of wholes, or operations, etc.) To show that what students already know about whole number guides them to understand what is true of fractions, he shows the relationship between arithemetic on the number line for whole numbers and arithmetic on the number line for fractions.

That means we need to already have mastery of the number line for whole numbers in order to make fractions clear. That mastery means fluency showing addition, subtraction, multiplication and division with remainder on the number line, as well as flency with properties of commutativity, associativity and distribution. The number line should reinforce all of what we already know about whole number arithmetic. It should not be new--we should use our mastery of whole number arithmetic to arrive at mastery of the number line, and then we'll use mastery of the number line for whole numbers to achieve understanding of fractions.

So let's begin:
The standard number line for non-negative numbers is a ray, beginning at Zero and extending to infinity. We show a piece of it here:

By convention, this ray extends to the right. The non negative integers, or whole numbers, are shown at the has marks. The space between 0 and 1 is a fixed length. This length is the same length as between 1 and 2, and any two consecutive numbers. We don't care what the length is, but we do like to recognize that it represents a UNIT. Later, this will matter more, but for now, just keep clear that the actual distance between the 0 and 1 is arbitrary; the issue is that that spacing is consistent for each consecutive number.

The biggest confusion with the number line is that it starts at Zero. This is probably odd for kids because we don't hold up our first finger and say "Zero!" But here, the First Tick Mark is the Zero. So we have to learn to go from counting on our fingers, where we sort of assume "zero means nothing", including no place/position, to an abstraction where Zero exists.

On the number line, Zero is our starting point. This is actually how we counted on our fingers, but we didn't think about it very often.

Now, we use Zero, the place, on our number line as our starting point, and count by keeping track of MOTIONS, or JUMPS from the starting point to the next tick mark. We don't count the fence posts; we count the motions from fence post to fence post.

Practicing with the number line until it's ALWAYS clear that there's 1 more fence post than motions between fence posts is critical.

Addition, then, looks like this: jumps to the right. (Again, this is convention.)
We add 4 and 3 by starting at Zero and making 4 jumps to the right. The first jump takes us to 1, then fourth to 4.

Then we make 3 more jumps to the right. The first of these takes us to 5. The last takes us to 7. We end up at 7.


Commutativity of addition should be obvious now: whether you made 4 jumps to the right and then 3 to the right, or 3 to the right and 4 to the right, you always ended up at 7. Associativity should be just as clear: 3 + (4 + 5) is the same as (3 + 4) + 5 for the same reason.

Subtraction is then defined as well. Subtracting a number from another number, you jump to the left (by convention.) 7 - 3: We start at zero, and take 7 jumps to the right to arrive at 7. Then to subtract 3 FROM 7, we make 3 jumps to the left. We end at 4.

Multiplication then is just a set of grouping of jumps. 4 X 10 is 4 Tens. That means, you learn to make a set of ten jumps, and then you call that grouping something like "a ten jump". And now you make a total, from 0, of 4 of these grouped jumps.

(Exercise: How do you do division with remainder? Answer: For X divided by Y, you start at zero and make grouped jumps of length Y. When you reach X, you stop. You count the number of grouped jumps you were able to complete, and the remainder is the number of singleton jumps left to reach X.)

The critical key no matter what the operations is to remember not to be counting the Fence Posts, or the hash marks between the numbers (inclusive or exclusively), just the jumps.

If the jumps are clear, then the next thing to realize was that you could have visualized those motions between numbers not as jumps but as continuous motions, where we moved along the number line progressively:


This is the simplest way to connect it back to fractions, because when we look at fractions, we now will consider the space between the hash marks.

Critical Concepts for Understanding Fractions

Hung Hsi Wu has written a document for the NMAP about fractions. This document "contains a detailed description of the most essential concepts and skills together with comments about the pitfalls in teaching them. What may distinguish this report from others of a similar nature is the careful attention given to the logical underpinning and inter-connections among these concepts and skills."

Basically, this is a document about what you need to know about fractions and how they work. This is how they should be understood. It is NOT a teacher's manual, and certainly not a student's textbook, but it is the mathematical basis of one, and it's a darn good start.

It is dense. Like a good math paper, there are no extra words to confuse. There is exactly as much precision and description but no more. That makes it slow reading if you are unfamiliar with the material.


Read it for yourself. Do you know everything it says? If not, does it help you identify your own lapses in understanding fractions? Does it help you to explain these concepts to your own youngster?

Over the next few days, I'll write more posts about this document, expanding on some of his writing. Perhaps we can make a KTM parent manual from this document with enough feedback and examples.

Math education IS mathematical engineering

Math education is mathematical engineering, so says Professor Hsu-Shieh Wu, professor of mathematics at UC Berkeley, topologist, gifted teacher, member National Mathematics Advisory Panel, etc. He is clear: this is not an analogy, but a definition.

Wu gave the plenary talk at the National Council of Teachers of Mathematics (NTCM) Annual Meeting in 2007. The comments here are taken from the slides of that talk (available here) and from recent papers he has written on the same subject, that of the relationship between mathematicians and math educators (see here). The errors or confusions should be considered due to me, rather than Wu. Consider the comments in italics to be straight from his talk or papers, and the rest to be from me.

Wu uses this definition of engineering: engineering is the customization of abstract scientific principles to satisfy human needs.


Mechanical engineering, then is about turning the laws of classical physics into pulleys, cranes, refrigerators, etc. It's how you build your cel phone or IPod so you can drop it without it breaking. Likewise, chemical engineerings is about turning chemistry into the plexiglas for your aquarium, the non toxic antibacterial cleanser you use on kid's toys, the gas you put in your car.

Mathematical engineering, then, is about customizing mathematics for use by students and teachers in K-12. That is the human need of mathematical engineering. Without the customization of math to students and teachers in K-12, math cannot be used by adults. It is as if you were unable to make any use of chemistry or physics. Math education, and its pedagogy, matters deeply. You cannot simply throw adult math at school age children.

Engineering gets better over time, with refinements, but the underlying principles don't change. Engineering must find its way between two principles: the inviolable scientific principle, and user friendliness of the end product.

Likewise, mathematical engineering must improve with better pedagogy, better adaptation to the students, etc. but it can not change violate the principles of mathematics: 1) precision, 2) definition, 3) coherence, and 4) reasoning, 5) purposefulness.

Precision means making clear, unambiguous mathematical statements. There is no unspoken context in math: you are expected to make clear what is known/given, and what is unknown.

Definition: concepts in math have specific definitions. A specific concept as a specific definition--one, and all others can be shown equivalent. But the structure of math demands a definition for all concepts.

Coherence: math builds on prior knowledge. Nothing comes out of the blue, but it unfolds from what is known already.

Reasoning: mathematics cannot proceed without reasoning. Reasoning must be illuminated.

Purposefulness: math is goal oriented. It solves specific problems.

Mathematical engineering, that is, math education, to date as lost these principles. While there is some pedagogy available for teachers, over and over again we see that the teaching of basic ideas (geometry, fractions, etc.) of math violate most or all of the above 5 principles. This is unworkable engineering: it is the equivalent of chemical engineering labs that don't know enough chemistry to create stable chemicals.

In other papers as well as this talk, Professor Wu elaborates on the failures of current pedagogy to address those 5 principles. He does this by showing the failure in the presentation of fractions, geometry, and in what he calls the Fundamental Assumption of School Mathematics. I will address the fractions presentation in another post.

For geometry, he states that the mathematics of Euclid and Hilbert goes from axioms to theorems to proofs in an un-user friendly way. However, the major school presentations either teach it this way, without addressing students' learning capacity, or they present it without definitions, theorems, and proofs, as if it is experimental geometry, and so it lacks precision, coherence, reasoning, etc.

For the Fundamental Assumption of School Math: mathematically, it is true that all arithmetic operations on fractions (i.e. rational numbers) can be extrapolated to work on the reals. Why this is true is nontrivial mathematics. School math is only about rationals, and then presentation of irrationals is done without any such claim or explanation that it can be done. The assumption is left unstated: a vioation of the precision, definition, reasoning, etc.

So mathematical engineering requires both mathematicians and math educators, just as chemical engineering requires engineers and chemists. The math educators are needed because they know the students, and what is needed by the students at their various ages. Math educators know what the school math curriculum is for a given level, if not how to present it. They, too, are familiar with what the maturity level of their students is. Mathematicians are needed because correct mathematics must be taught at each of these levels, even though what is known at each level is different. That means a variety of correct explanations must be made available. That requires deep subject knowledge--deep enough to understand what's true and correct from a variety of view points. And in truth, these issues come together as we attempt to find the best approach for each student: you need both pedagogy and mathematics in order to reach students and still teach them math that adhered to the above 5 principles.