Several years ago, I was working with a class of fourth and fifth graders. Their teacher had begun a unit on fractions and was interested in connecting fractions to real-world contexts. “No problem,” I told her.

Our plan was that I would teach a lesson, she would observe, and then we’d revisit it. I’d focus on talking with students about naming fractional parts, the standard symbolism of fractions, and equivalence.

**My first real-world context: a six-pack of water**

I showed the class the six-pack I had brought to class and talked about one bottle being 1/6 of the six-pack, two bottles being 2/6, three bottles being 3/6, and so on up to 6/6 being the same as the whole six-pack. The students seemed comfortable with this, and I wrote the fractions on the board:

*1/6 2/6 3/6 4/6 5/6 6/6*

We also talked about three bottles being one-half of the six-pack, and that 3/6 and 1/2 were equivalent fractions because they both described the same amount of the six pack. I recorded this:

*3/6 = 1/2*

I asked what fraction of the six-pack would be gone after I drank four of the bottles and they answered 4/6 easily. I represented this numerically:

*1/6 + 1/6 + 1/6 + 1/6 = 4/6 *

I asked what fraction of the six-pack would be left after I drank four bottles, and they answered 2/6 easily. I represented this numerically with two equations:

*6/6 – 4/6 = 2/6
*

*1 – 4/6 = 2/6***My second real-world context: a box of 12 pencils**

I continued with a different context—a box of 12 pencils. We talked about one pencil being 1/12 of the box, two pencils being 2/12, three pencils being 3/12, and so on. I wrote these fractions on the board:

*1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12 12/12*

The pencil box gave us a way to talk about another equivalent fraction for 1/2, this time 6/12. And I talked about 12/12 representing the pencils in the whole box:

*6/12 = 1/2
*

*12/12 = 1*I asked, “If I give a pencil to each of five students, what fraction of the pencils would I have given away?” They answered easily and I recorded numerically:

*1/12 + 1/12 + 1/12 + 1/12 + 1/12 = 5/12*

I asked if 5/12 represented more or less than half the box, and they agreed that it was less than half. I recorded again:

*5/12 < 1/2 *

I asked a few other questions, recording numerically for each. All this was going fine.

**Then I hit a snag
**The students in this class sat in small groups, and I next called the students’ attention to a table where two boys and one girl were seated. I asked them what fractional part of the students at the table were girls. Hands shot up and I had them say the fraction in unison in a whisper voice—one-third. I wrote

**on the board.**

*1/3*Brad noticed that the table next to his also had two boys and one girl sitting at it. Claudia commented, “So if you put the two tables together, then 2/6 would be girls.”

Addison’s hand shot up. “Can I come up and write that in fractions?” he asked. I agreed. Addison came up and wrote on the board:

*1/3 + 1/3 = 2/6*

I was stunned. Addison was correct that 2/6 of the students at the two tables were girls. But the addition equation that Addison wrote wasn’t correct. It’s every teacher’s nightmare when students combine the numerators and denominators to add fractions and think that adding 1/3 and 1/3, for example, gives an answer 2/6. But I didn’t think that Addison had applied that incorrect procedure. I wasn’t sure exactly what he was thinking.

To buy some time, I asked Addison to explain what he had written. He said, “One out of three at Brad’s table is a girl, so that’s one-third. And it’s the same for Margaux’s table. So, if you put them together, then two out of all six kids are girls, and that’s two-sixths.” The rest of the students nodded.

They were all pleased. I was a wreck.

So much for buying some time.

**It’s hard to think and teach at the same time!
**I stood quietly and thought for a moment about what to do next.

To fill the quiet, I said to the class, “When thinking about fractions, it’s important to keep your attention on what the whole is.”

They nodded politely.

After thinking some more, I returned to the context of the two tables of students. I said to Addison, “I see that you’re thinking about the two tables together.” He nodded. “So, the group of students at the two tables together has six students.” He nodded again. “Then Brad, Samantha, Jack, Margaux, Robbie, and Max, are each 1/6 of that group, just as each bottle of water is 1/6 of the whole six-pack.” Another nod. And because 1/6 + 1/6 equals 2/6, it makes sense to me that 2/6 of that group of six are girls.” I wrote on the board:

*1/6 + 1/6 = 2/6*

None of the students seemed concerned that 1/3 + 1/3, as Addison had written, seemed to produce the same answer as 1/6 + 1/6, as I had written. Now I was breaking out into a sweat.

**I tried again to explain
**“Let’s look at just one of the tables,” I suggested. “There are three students—Brad, Samantha, and Jack. What fraction of the table does Brad represent?” The students answered 1/3 easily. “And what fraction does Jack represent?” They answered 1/3 again. “And what fraction of the table are boys?” They answered 2/3. I wrote on the board, underneath what Addison had written:

*1/3 + 1/3 = 2/6*

*1/3 + 1/3 = 2/3*“Hey,” Addison said. “You got a different answer to 1/3 plus 1/3.”

**What would you do now????? **

Ahhh, this reminds me of your old interview with David. It was something like this:

If each friend gets 1 piece of a small pizza, divided into 3 equal pieces.

What would be the share for two friends?

David answered 2/6 at one point in the interview and answered 2/3 at another point. His reasoning was correct both times. 2/6 is right if the “whole” is six slices of pizza. 2/3 is right if the “whole” is a single pizza.

It is all about the whole.

What would I do? I do not know – perhaps try to find a not-as-clumsy as the following way to rewrite the equations with units?

1/3 of the seats available at one table + 1/3 of the seats available at one table = 2/3 of the seats available at one table

1/3 of the seats available at one table + 1/3 of the seats available at one table = 2/6 of of the seats available at two tables

Yes, keep the focus on what’s the whole. It’s hard, for the kids to figure out and for me as a teacher to feel as if I’m shooting from the hip searching for ways to help.

When the whole was 1 table of 3, they were really adding 1/3 out of each whole, so 1/3+1/3= 2/3. When they considered all six students in both tables the whole, then they had 1/6+1/6=2/6.

I just finished Fractions AVMR training and it has blown my mind to understand all that we have traditionally missed in math understandings!

So what did you do?

At the time, I let it go so I could think and regroup myself. There was more than fractions to think about: Is it possible for both equations to be true: 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6. This was a long discussion that kept their interest, in which I learned that many (most?) of the students didn’t think of fractions as numbers. That was another hurdle. I kept going back to what we know about adding whole numbers. Then I took another approach, once I was sure that they understood that 1/3 and 2/6 are equivalent, so how can I add a number to itself and wind up with a sum that’s the same as one of them? In all of these discussions, students would change their thinking. We looked at adding on a number line. I used pattern blocks to explore the same problem. I kept talking about keeping our attention on what was 1 whole. My, this all takes time, and the time is important for students to develop, cement, and extend their understanding. What I didn’t do, that I’ve been thinking about now, is to make it part of a writing workshop on persuasive writing and have them choose a conjecture and write. I think I could spend most of the year on this with students.

This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding.

Thanks for your point about ratios. The complexity of fractions is important and often astonishing.

Thanks! This is a great illustration of how students try to apply fraction algorithms to problems which are really about combining ratios.

The bottle of water context is definitely a problem in which we add fractions, hence the denominator stays the same. The ratio of girls sitting at a desk compared to total students sitting at a desk is a combining ratios problem. Thus it makes perfect sense in the case of the tables to look at the new whole of 6 students, of which 2 are girls.

In my courses for preservice K-8 teachers, we have had wonderful conversations about this very notion, and people are always intrigued about the ways in which fractions and ratios sometimes operate the same, and other times, operate differently.

Thanks for your comment. Do you think the blog is appropriate for preservice teachers to consider? I’m thinking about that (I’m soon to be a guest teacher in Ruth Cossey’s class. Maybe they should read the blog in advance. Just thinking.

Yes! I think your blog would be a fantastic resource for preservice teachers, as it may give them a sense of the types of things that practicing teachers will encounter in the classroom.

As you are aware, many of our elementary preservice teachers are developing their mathematics identities and confidence, and this blog may allay their fears of confronting interesting mathematical questions during class, where they will need to think in the midst of teaching. I try to assure them that this happens all the time to us as teachers, and that it’s totally fine to say, “Wow, that’s a great question! Let me think about this!”…and even get back to their students the next day with an answer.

I hope some people use in with preservice teachers and report back. Thanks.

I think it was a good thing one of the students didn’t decide to calculate that 2/3 of the students at each table were boys and add 2/3 + 2/3 = 4/6, because that would have totally reenforced their thinking. I wonder if there had been a table of all girls or all boys that could have served as a model. That way it could have been shown that by adding the table of 2 boys and 1 girl together with a table of 3 boys would not have produced 1/3 as an answer. Hmm tricky situation.

That would have been good, and possible. It’s a reminder of how hard it is to think in the midst of teaching. Now I’m wondering about presenting the conundrum to another class of 4th-5th graders and engaging them in the discussion. Maybe even give them my blog to read. Might be good for a problem-solving lesson. Stay tuned.

I have seen a similar issue arise when students draw models. For example, I have seen students draw 1/3 on one bar model, and 1/3 on another and call the sum 2/6 ad they have unwittingly change the whole from 1 bar to two. With a fraction of a set context I suspect it would be even easier to not realize that the whole had changed.

See Ruth Cossey’s comment about students’ confusion. So common.

Agreed: it’s all about the whole.

I would use bar models to show how the whole changes.

First, draw one bar (table) with 3 students inside; label the bar one whole, star one student and label that student as one third of that whole. Do the same thing beside the first bar model, again showing the bar as one whole, starring one student and labeling that student as one third.

Then, push the two bar models together (I’d use a Smart Board) to show a new whole: the two bars together with 6 students in the one bar is labeled as the new whole.

Point out that this is a NEW whole, with a different # of pieces. Now they would see the 2 starred students in the one new bar made up of 6 total students, so the “old” 1/3 student becomes the “new” 1/6 student once the whole changes. This is like a name change, once the size of the whole changes.

You can also show them that by cutting each “student” in half, you would get an equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, not two 1/3’s = 2/6.

This is a big idea that can be confusing!

Thanks for your thinking about how to structure a lesson. Planning and preparing are so important. I haven’t used bar models much and need to think about this.

Fascinating! This blog post is so important! It shows how often students make the context to symbol link in maths but miss the conceptual/visual understanding (in this case the proportion of 2/3 versus 2/6), and how crucial it is to sometimes explore a concept in multiple ways, several times to really find out if students have ‘got it’. I would have left it there then returned to revisit the problem using different materials (e.g fraction pieces) and prompt them to find out where the error came about.

PS- I use bar modeling for everything as it gives a clear visual representation of quantity and proportion, which is where maths tends to fall down for most students when introducing more complex concepts.

I would push on the meaning of the denominator (how many equal pieces are in each whole) and how it’s different for each situation…tying the expressions to the visual.

I also think that tying this to an area model and number line can be a huge help.

Fractions are a number in and of themselves which is seen on the number line. Ratios, however, are separate numbers, and we need at least 2 number lines to show them.

I’ve been working with a group of 6th-grade teachers about pushing on this learning with their students so that they know the difference between ratios and fractions.

Thanks for describing the difference between fractions and ratios so clearly. Now I’m thinking that with 4th and 5th graders, the learning should focus on building a firm foundation of fractions that they can bring to thinking about ratios. Oh, I wish I still had this class available to me now.

I may not have thought of this “in the moment” but looking at it now, with time to think….This is more about the “ratios” which is not something usually taught in 4th/5th, but it could be a teachable moment…When there are different wholes the “ratio” of girls to boys is the same, but the fractions can’t be added if you are using two different “wholes”. You could add a few examples of simple ratios and then get back to the rules of fractions 🙂

Oh, my, the many decisions we have to make when teaching. Yes, it seems like a “teachable moment” to move into ratios, but I’m wondering if that’s a good route or continuing to build a foundation of understanding about fractions. I don’t think there’s a right answer — I’m just wondering.

I wonder what the kids would say if they compared the two equations. If they compared the water bottle equation 1/6 + 1/6 = 2/6 and the table equation 1/3 + 1/3 = 2/6.

If I asked “Are 1/6 and 1/6 equal?” and “Are 1/3 and 1/3 equal?” By their logic, 1/3 and 1/6 are also equivalent.

Then if we posed the question 1/3 + 1/6 = ? and asked them to model out 1/3 and 1/6 with some kind of visual, I wonder if they’d realize they aren’t the same and can’t add up to the same value.

I like this idea of getting them to think about water bottles and students at tables together. Thanks!

What about going back to counting by unit fractions. If you are counting by 1’s, it’s like adding 1 each time, so you say 1, 2, 3, 4, ….. If you are counting by one thirds, it’s like you are adding 1/3 each time, so one 1/3, two 1/3s etc or 1/3, 2/3. 3/3, 4/3 etc. This might help them see that they can’t start at 1/3 and get to 2/6 by counting or adding on.

Along with that I would keep the focus on the whole. And I wonder whether using an area model of fractions might help and then connecting that back to the set model.

Or representing 1/3 and 2/6 on a numberline – how can 1/3 + 1/3 = 2/6 if they both are placed on the same spot on a number line?

I am co-teaching in classes right now and students are exploring and building understanding of representing fractions and naming fractions from representations. There have been interesting discussions when we have talked about the fraction of a whole hexagon. If a hexagon is covered by 4 triangles and 1 rhombus, is the fraction represented by the triangles 4/6 or 4/5 (given that the area of the rhombus is twice that of the triangle). If we are talking about the area of the hexagon then it is 4/6, but if we are talking about the number of blocks used, then 4 out of 5 or 4/5 are triangles. Challenging concepts for our students to struggle and learn and make meaning.

Thanks, Kit. Your suggestion in the first paragraph would have been useful (I think) in getting Addison to think about why 1/3 + 1/3 couldn’t be 2/6, but rather should be 1/3. And certainly work with Pattern Blocks is appropriate here. Thanks for giving me more to think about.

I resonate with your last paragraph. I had some students build 1/2 + 1/4 and then another student build 1/4 + 1/4 + 1/4. They kids could visually see they were the same thing but didn’t know how to say it. We spoke about how when you want to describe an image based on a single fraction then you have to create equal parts. This was a great discussion in a grade 3 class, I was not confident they totally got it. I’m thinking I’ll have design a lesson around this challenge. Any thoughts?

My go-to is using Fraction Kits. Let me know if you’d like more info.

By fraction kits are you referring to Manipulatives? We definite use manipulatives quite a bit. I am currently working in a school for non-typical learners and I attempt to make every lesson spatial with minimal worksheets unless they are organizational in nature.

Yes, fraction kits are a concrete way for students to interact with fractions, actually a length model. I give students strips of different colors, have them leave one uncut, then cut and label another in half, another in fourths, then eighths, then sixteenths. I think the cutting and labeling is important and valuable for students to do, so they create their own fraction kits. Later they make thirds, sixths, and twelfths. They play lots of games. And I use the kits to help them think about fractions on a number line, which I blogged about here: I’ve written about fraction kits in the Teaching Arithmetic series: Lessons for Introducing Fractions, a Math Solutions resource.

If counting the total of students at two tables the denominator must be six, so 1/6 plus 1/6 equals 2/6 or 1/3.

Ah, now to convince students. Or, better yet, to have students convince themselves.

I love that you posted this- such an easy dilemma to find yourself in! To answer the question, what would you do: I think I would stress that we were talking about groups, just like the 6 pack of water came in a group, we were looking at a group sitting at a table. Adding in the other table is another group. I would also look at similar situations that involve groups as a whole.

I am curious if anyone else has difficulty using the vocabulary of one whole vs one. We are just finishing up our first unit on fractions and some of the students seem confused by the word whole and don’t see it as equal to one. I have become very conscious about how I label one and/or a whole and/or a group that has all the pieces so the numerator and the denominator are the same.

I worry that using “whole” or “one whole” may add to the confusion about whether or not fractions are numbers, which confuses students. I’ll have to think more about this.

During number talks on fractions, I’ve been asking my students, “Okay, before we begin what do we need to decide as a group?” Then they chorus back, “How are we going to talk about one?” I use both terms, whole and one, but students don’t seem to. They have definitely latched on the term one rather than the whole. Now, I’m wondering, is there an issue with talking about it as one rather than a whole?

Read Phil Daro’s comment to this blog post. He explains well, I think.

I think the point that has to be made is that 1/3 is no longer really 1/3 when you introduce the second table. Students must realize that the 1 girl seated at table 1 is really now 1/6 of the entire group. Likewise, the individual girl at table 2 is also 1/6 of the entire group. Therefore, 1/6 + 1/6 = 2/6. I would draw a picture and use a bar diagram to help contextualize the problem. (I’ve drawn it out on a Google Doc., but can’t attach it. I’d be happy to send it to you if you’d like it.)

I’ve drawn pictures as well. Sometimes, it seems that we work soooo hard when we teach and still it’s difficult to connect. I guess it’s all about the students making sense, and they often respond to different models and drawings. Onward.

I would challenge the students to think why this is not correct, first. Growth Mindset 🙂

I would change the representations to shapes and have 2 circles cut into thirds. Another circle cut into sixths.Then take out 1/3 of each circle and place them over the circle cut into sixths. This would show that they are larger than 2/6. Then I would discuss about the whole and not just adding parts of different wholes together.

I would also thank the student for challenging me to prove my reasoning with fractions.

Thanks for the ideas. It’s interesting to think about shifting from parts of wholes (like circles) and parts of sets (like bottles of water or pencils). I have lots more to think about.

I know exactly what this feels like! I’ve had similar situations when teaching fractions (decimals and percents) and it happens every year. I stay on this topic for a long time in number talks to help the students build their language for it. There are a few confusing things going on at the same time here since they are just learning fraction concepts and the students are jumping ahead to try and add them.

At that point in the lesson, I might cut two circles into thirds and have the students at the table distribute them between each other. Emphasize that each student represents ⅓ of their table, then they could try adding them in different ways or calling different fractions to stand up (If you add the two girls, you get ⅔, but if you add all the students you will have 6/3). This could lead to a lot of fun questioning for you and discussion for the students.

Thanks for the suggestions, and for your understanding of my situation.

Great discussion

I see two issues:

1. part/whole model is mathematically immature and won’t support number sense that fractions are numbers. Part/whole is a type of context where fractions are useful; it is not a definition of fraction. A fraction is a number.

The number 3 is “3 of 1”, where 1 is whatever we are counting or measuring: the unit. A student can count whatever he or she wants; she is the agent. Likewise, 1/3 is “1/3 of 1” where “of 1” means exactly the same as it does in “3 of 1” . 1/3 is a number with reference to the number 1.

In many situations, there is more than one 1: we might count miles and also hours, dollars and also pounds of potatoes. We keep track by explicitly writing the “units,” that is, the ones we are counting. But not all numbers have concrete units.

We can do arithmetic where the unit is an abstract 1. But we still have to be clear what 1 we refer to. This is pretty common with fractions. 2/3 means 2 of 1/3; that is, we are counting the unit fraction 1/3. At the same time, 1/3 is 1/3 of 1. So 2/3 literally means 2 of 1/3 of 1. Yikes.

In the girls at the tables, 1 student is 1/3 of 1 tableful of students. What are we counting, students or tablefuls of students ? Both. Then we add another tableful so we have 2 tablefuls. And then we want to stop counting tablefuls and count everyone as a single group, this defining a new 1 and erasing an old 1. We do this behind the curtain without being explicit. Yikes again.

1/3 tableful + 1/3 tableful = 2 times 1/3 tableful = 2/3 tableful OR 2/6 of 2 tablefuls

2. The other issue is that the girls at the table is also a ratio. The ratio of girls to students is 1:3 at each table. If I have 1cup sugar and 3 cups water, and I add 1 cup sugar and 3 cups water, I get the same sweetness from 2 cups sugar and 6 cups water. We add ratios all the time: 1/3 + 1/3 = 2/6 as a ratio by adding the numerators and the denominators. It is a shame those ratios look just like fractions.

What if the second table had 1 girl out of 4 students? Then the new ratio would be 2 girls : 7 students.

Why can’t I write that as 1/3 + 1/4 = 2/7? Because fractions are numbers. Ratios are pairs of Numbers. A ratio has a unit rate which is the fraction of 1 quantity that pairs with 1 of the other. The ratio 2 girls : 7 students has a unit rate of 2/7 girl per 1 student. Sometimes the unit rate associated with a ratio is called the value of the ratio. Makes more sense if it’s 2/3 cup sugar per 1 cup of flour.

Notice in 1/3 + 1/4 = 2/7 there are three different units: thirds, fourths and sevenths. Addition does not allow adding different units. That’s why we make equivalent fractions with the same units to them.

4/12 + 3/12 = 7/12.

When we do this , however, what are saying about girls at tables? We are saying that the girls at the two tables combined are 7/12 of 1 table , but we changed the definition of table: our new table has 12 places which are shared equally among how many sit there. If 3 students, each gets 4 places and so. The number of places assigned to girls at two tables is 7, and each place is 1/12 of 1 table.

In conclusion, I am pretty sure the sources of confusion in this classroom arise from our own confusing of fractions with part/whole and with ratios. The student was correct if you grant him what his numbers referred to, that is, what he counted as 1 in each instance.

Thanks, Phil, for taking the time to write such a detailed response. You and I have talked about these issues before, and each time I confront them, my thinking gets clearer. It’s a reminder of why it’s so important, as a teacher, to really understand the math we’re trying to teach. And, in this particular case, to think not only about fractions but also in the context of ratios. Onward.

This is where I would ask, “How did we get two different answers? What’s going on?” It would open up the debate to the class, and usually they would be able to wrestle their way to an understanding that the wholes have to be the same.

Throughout, I would give hints as needed, such as, “One of them is wrong. Can you figure out why?” and “This problem is saying 1/3 of the table plus 1/3 of the table equals 2/3 of the table. What is this one saying? Why doesn’t that work?”

These seem like productive questions to ask. Thanks.

I love the honesty of your post, “I was a wreck.” We’ve all been there when the unexpected happens. You take a deep breath and muster the confidence needed to jump down a rabbit hole that could be a great teaching moment. And wow! What a great moment. I think Kelli nailed the big question, “What’s going on?”

My first reaction would be to ask, “What would happen if we moved things around a bit?” Then, I would ask the girl at the second table to switch places with a boy at the first table. “What fractions do you see now?”

I’m a fan of conducting math congresses, which I would convene to discuss the different representations the children saw. Sometimes children don’t come to an agreed correct answer, so we have an “open question,” which is exactly what happens with mathematicians. Then, we have to explore new models for more evidence. But the math congress rule I have is: open questions cannot be resolved until we know why the incorrect answers are wrong AND why the correct answers are right. IMHO, we learn more from our wrong answers. Rewarding mistakes and embracing productive struggle, right?

Thanks for this response. I love the idea of having students switch places, and then try and deal with representing the different configurations. And I love your description of conducting math congresses.

Oh the fractions. They are the thorn in my side, but I’ve learned to love them. Since fractions such a BIG part of the Nebraska State Assessment (NeSA), I give my students one fraction every other day starting with the first day of school. We take that one fraction, learn its parts, convert to decimal, percent, etc. They were even shocked to learn that the numerator is also the dividend and the bar means to divide. So by the time I have to teach the fraction unit, my students have an excellent handle on them.

The “whole” is a whole in the head; it’s not defined by mathematics but by the thinker…the problem solver. It’s pretty common for there to be different wholes inside one problem. That’s one reason I like to grow kids from “whole” thinking to unit thinking. It’s easier to think about different units in the same problem than it is to think about different wholes.

There are three whole thirds in one whole one. 1/3 of 1 is 1/6 of 2 . But 1/3 of the whole is 1/6 of another whole?

What I’m so impressed with, is how I have been using your outstanding resources for the past 20 years and you are still very much learning along with the rest of us. Thank you for your inquirying mind and continually digging deeper. You continue to be one of my female math heroes.

We have spent a great deal of time talking about the different meanings of the a fraction..part/whole (1/3 as 1 out of 3 equal parts), division (1 whole divided into 3 equal pieces),units of measurement (1 unit of one third, a number ( between 0 and 1). I wonder if this would be best explained through the idea of division? 1 whole divided into 3 equal parts would create a part called 1/3. Another whole divided into 3 equal parts would also create a part that is 1/3 of the whole. If you then separately drew 2 wholes divided into 6 equal parts, you once again create parts that are 1/3 of a whole. How could 2 of these 1/3 parts be equal to 1 of them ?

I think the link between fractions and division is really important. My struggle is to think about how best to give students access to thinking this way. And to be able to think about fractions in their many forms and purposes. I’m still mulling. Thanks for adding to the mix.

First of all, thank you, Marilyn, for always writing about teaching with so much honesty and continuing curiosity. I have always admired how much faith you have in kids, and how your instructional choices always reflect that confidence in their ability to figure things out when we let then talk and work together.

As a middle school teacher, I’ve wondered for quite a while why students have a much easier time accessing their number sense and intuition around ratios than fractions. In reading your post and the many responses, I realized that ratios are never naked numbers – they always have units built in, so that the numbers do not need to be abstract. In contrast, it seems like fractions almost never have the unit attached, and so students are not used to asking themselves “1/3 of *what*?” as a way of helping make sense of fraction problems. This is why I found Kelli Pearson’s suggestion above intriguing.

I’ve never had the chance to think about how to support students in realizing that fractions are numbers. In middle school, Like Myra Hudson above, I ask students to think about fractions, decimals and percentages as different representations of the same thing, but I’m not sure that helps them see fractions as numbers.

You’re right that I’m committed to giving students support and opportunities to figure things out for themselves. But sometimes it’s really hard and takes a really long time, so the pressure of dealing with all we need to teach raises my anxiety. What you write about ratios having a natural link to contexts makes sense. Students always connect new learning to something they already know, and naked numbers don’t provide that support.

Although the fraction notation is a very powerful way of solving division problems (2 divided by 3 is 2/3, where the question ‘provides the answer’) the fraction notation is very opaque. Students in many countries have great difficulty with the notation – they do not see 2/3 as one number, they typically see it as 2 whole numbers. To make the notation even more opaque we describe fractions with ordinal names (eg thirds).

One thing that I have learnt (albeit very slowly) is that students cannot use what they don’t have. They cannot use area models of fractions if they don’t understand area. And they cannot use the discrete models of fractions if they cannot identify and hold on to the ‘unit whole’.

The class discussion got on to re-unitising the whole – a very difficult thing if you readily lose sight of the whole. I have often used your fraction strips where we spend time being certain that we are always using the same unit whole.

I understand the value of your blog is the discussion it generates, so I will leave my comments at this point.

Thanks, Peter. I agree that fraction notation adds to the confusion. And my experience has been that when students use some models for fractions, like area, when their understanding of area is fragile, I feel as if I’m fighting too many battles. It’s not that I want to simplify their learning, but I want students to have enough time and experiences to make sense of fractions in all their complexity. It’s tough and it takes time. When I told the class of fourth graders I was working with the year before last that my grandson, then a sixth grader, told me that fractions weren’t really numbers. The students in the class responded by telling me that my grandson must be “really smart.” I responded, “I think he’s smart, but he wasn’t right in this case.” And then we talked about fractions as numbers. I feel that each time I work with a class on fractions, I get better at predicting students’ soft spots and helping them make sense. But it’s never a quick process.

Marilyn, such a great post and true reflection of how hard it is to try to reconcile students’ thoughts, your own teacher thoughts, and best teaching move at that point, all within the same class period! Even the best planning puts us in those situations all of the time, right?

While I know this wasn’t a fraction of a set problem, I can see how students may be seeing it that way. I have had many conversations with teachers about fraction of sets (such as the water bottles or pencils) in the earlier grades because I would see that as something I taught in 5th grade when students begin multiplication of fractions. My only thought at that time about why I felt this was a 5th grade thing over let’s say a 3rd grade thing was because of notation because even when the teacher is seeing 4/6 of the pack of bottles, students are seeing 4 bottles or 4/6 of 6 or 4/6×6.

After reading this thread, I am thinking less about how students would notate their thinking and more about the confusion it creates in defining a fraction as a number because of the confusion when attending to the unit.

I had to read Phil’s comments a couple of times because there was so much packed in each statement, but they were so helpful in my own thinking around this. These two parts I found particularly helpful and so well-articulated:

1) “The number 3 is “3 of 1”, where 1 is whatever we are counting or measuring: the unit. A student can count whatever he or she wants; she is the agent. Likewise, 1/3 is “1/3 of 1” where “of 1” means exactly the same as it does in “3 of 1” . 1/3 is a number with reference to the number 1.”

This makes me always want to ask each student what they are counting as 1, see how their responses are the same or different, and in turn how that impacts their response to a question such as 1/3+1/3.

2) “In the girls at the tables, 1 student is 1/3 of 1 tableful of students. What are we counting, students or tablefuls of students ? Both. Then we add another tableful so we have 2 tablefuls. And then we want to stop counting tablefuls and count everyone as a single group, this defining a new 1 and erasing an old 1. We do this behind the curtain without being explicit. Yikes again. 1/3 tableful + 1/3 tableful = 2 times 1/3 tableful = 2/3 tableful OR 2/6 of 2 tablefuls”

Yikes is right! I am still re-reading this paragraph and loving it more each time I do.

Thank you so much for putting this out there and cannot wait to hear about someone using this with pre-service teachers!

-Kristin

Thanks for your detailed comment. Phil has the extraordinary ability to pack a lot of truth into a concise description. He and I have talked about these ideas, and I’m still grappling with how to integrate them into my classroom teaching and make decisions, both in planning and in executing lessons. I’m still mulling over whether to try this with pre-service teachers and, if so, how to do so. Stay tuned. If I do it, I’ll do it before NCTM, so we can talk then, yes?

Great post and reflections.

I would talk about not mixing different units, if you’re adding thirds of the small tables, you can’t get an output of sixths of the combined table.

Thanks. The issue of the small tables first and then the combined table is important, for sure.

I think what the student was doing, intuitively, was taking an average. (1/3 + 1/3)/2 does equal 2/6 or 1/3. If you had one girl at a table and two girls at another, the equation would be (1/3 + 2/3)/2 = 3/6 = 1/2, and if you had one girl at a table and all girls at another, 2/3 = 4/6 = (1/3 + 3/3)/2.

The problem is trying to explain why this works, and why the 1/3 + 1/3 = 2/6 equation is a special case where the numbers seem to work, but that’s not been written correctly.

Hmmm, I’ll have to think more about this. It’s tough at times to follow someone else’s reasoning.

Thanks for this great discussion and all the different perspectives! I agree that fractions are all about the whole.

I think that Addison’s problem here is that the one girl at first table of three was first identified as a ‘third’, so he only thought of one table group of 3 as being the whole. However, he hadn’t understood that when considering the students from two tables, these need to be combined to became a larger, SINGLE set, and that the whole set/group changes from three to six students, so that each girl is now a ‘sixth’ of the whole group and no longer a ‘third’ of their respective table groups.

What I might do at this point, is to go back and have students re-identify the first girl as a THIRD, 1 ‘out of’ (the slash in the fraction) her table group of 3 students. If she is ⅓ of her small group, what fraction of her same group are the boys? (⅔) Following this, I would ask a boy from the second table to join their group, having students now identify the same girl as a ‘fourth’ of the new group of four. Then, I would continue and add the last boy from the second table, so that the girl is now one fifth of the now larger group. I want students to understand that a single ‘unit’ (in this case, one girl) needs to be renamed as the whole changes.

Finally, I would ask the girl from the second table to join so that the whole group now has six. What fraction is the first girl? (1/6) What fraction is the other girl? (1/6) What fraction is the two girls out of the whole group? So then, count the sixths: “one-sixth”, ” two/sixths”. We can also write this as an addition statement: 1/6 (first girl) + 1/6 (second girl) = 2/6 of the whole set of six students. What fraction of students have shoes with laces? What fraction are wearing read? If you then add two or three more students (either girls or boys), can the students identify what fraction each student represents? What fraction speak another language other than English? What if the whole set/group is the whole class?

The confusing (but interesting) feature about this problem is that the ‘whole’ set of six students is comprised of two equal and smaller groups (the two table groups of 3). So, it’s important to help students understand what the true ‘whole’ set really is for any given context.

Oooh, I wish I had thought of this at the time. It seems like it would have been really helpful. Ah, hindsight.

Ah yes, I want to add that I love your resources and ideas that I have used since the 1990s. They have inspired me to teach, not just math, but all areas of the curriculum with understanding and to have children ask questions and to make sense of what they are learning. Thank you, Marilyn!

Thanks for this extra message. You’re very, very welcome!

I would tell her she found the mediant and maybe even show her the symbol (plus sign inside a circle). Then explain how that is different from fraction addition.

I think it might be helpful to introduce the mediant concept (and “Simpson’s paradox”) at early stages of education, even if one doesn’t want to use the vocabulary. Explain to the kids what it *does* represent, then they might be less confused later.

Thanks for a great blog.

Wow, how did I get this far in math without learning about mediants or Simpson’s paradox! About mediants, I checked online and read the Wikipedia entry() and also an entry from the Math Forum (). About Simpson’s paradox, I watched this You Tube (). I’ve got my work cut out for me. Thanks for the suggestions.

You might also take a look at this NCTM article by Eric McDowell:

https://www.nctm.org/Publications/Mathematics-Teacher/2016/Vol110/Issue1/Mediants-Make-(Number)-Sense-of-Fraction-Foibles/

Admitting that I sometimes use the concept as a shortcut when comparing fractions (e.g., which is larger, 17/32 or 18/33?).

I think that starting fractions with discrete models and the part/whole notion of fraction sets them up for these kinds of problems. In a discrete model, it is difficult to explain why it matters so much what we call the whole, especially before operations on fractions appear. For a student who is saying that 1/3 means “1 out of 3,” I don’t know how to convey that they are correct, but heading for trouble.

Starting with a continuous, linear model, automatically emphasizes what the whole is. It is easier to give students meaningful experiences with fractions that are greater than 1 using a linear model. And of course the connection to the number line and the crucial goal of students understanding that fractions are numbers that Phil described is greatly facilitated by a linear model and transitioning to a number line.

I am teaching a course for pre-service teachers now, and as luck would have it I asked your question on a recent homework assignment. I described a child who used a discrete model to add 2/5 + 2/5 and came up with 4/10 as the answer. I asked my students to describe a conversation that they might have with the child that would allow the child to figure out what went wrong with their process. This is a challenging problem.

My students who used language like “2 out of 5” had the most difficulty with this problem – how does the child recognize that it matters that the whole changed during the middle of the student’s calculation? The best they could do was to show that the answer 4/10 makes no sense. My students who used continuous models in their explanations found this much easier to handle, because changing the whole mid-problem is more clearly not sensible – if you have 1/3 of a candy bar, and get another 1/3 of a candy bar … .

Students should certainly have a full range of experiences with fractions, using discrete and continuous models, but the more experienced I become, the more I believe in the value of using a linear model as the starting paradigm, the place where students become comfortable about the meaning of fractions.

Thanks, Scott. I’m going to teach a preservice course the week after next (it’s Ruth Cossey’s class at Mills), and we are going to use the blog and the comments for most of the content for the session. I am curious to see what I learn. I’ll report back.

Back in the day, we used Cuisenaire Rods to help with this kind of situation. The concept of what is the whole in the problem determines what parts the fraction takes. With Cuisenaire Rods, the colors were fixed, so depending on what you defined as 1 would determine the fraction or whole number another color had. So if Dark Green was 1, then red was 1/3. Yet if I defined brown as 1, that same red would now be 1/4. If red was 1 then white becomes 1/2 and dark green became 3.

We would spend a lot of time talking about defining “1” first to determine what fraction or whole number equivalent another color was.

Doing this activity first might have made it easier for the class in your example to understand that by adding another table to your problem, you have re-defined what 1 stands for.

Alas, I don’t believe teachers nowadays have any concept of what Cuisenaire Rods were or how to use them. They mostly use fraction pieces that have fixed shapes with a fixed definition of 1.

All makes sense. I’m saddened that Cuisenaire rods aren’t used much these days. They have so much value. Ah, me.

Funny that this should come up right after I was looking at my NCAA brackets. After the round of the Sweet 16, I had 1/8 teams left on one side and 2/8 teams alive on the right side. My brain knew that I had 3/16 teams still alive, but looking at the fractions I had penciled in along the bottom, it looked like 1/8 + 2/8 should add up to 3/8. (Sorry to say that I now have only 1 team left in the Final Four, hence 1//4) .

Thank you, Marilyn, for describing this scenario. It’s so easy to see how this misconception could occur. I was picturing how a student might form 1/3 (as a bar model or with color tiles) and another 1/3 (in a separate model) and then push them together (using understanding about adding with whole numbers to combine them). This would certainly support their point that 1/3 + 1/3 = 2/6. I think I honestly would also have had these thoughts, “Wow, that kind of makes sense – but I know it’s not right – how can I explain this???”

After reading all of the other responses so far, I think I might approach this with 4th or 5th graders by asking them to count by one-thirds: 1/3, 2/3, 3/3, etc. AND I would combine it with the fact that once we put them together we no longer have thirds, but sixths – which would change the problem to 1/6 + 1/6 = 2/6.

Another reason it was great of you to post this is that this is likely to happen to any of us, experienced or not! I think I will forever be alert to this scenario. With the right group of students, it might be worthy to present this problem to them to see if they can adequately explain it!! I always believe that knowing what students’ misconceptions might be should drive part of our lesson planning to anticipate (and hopefully head off) what problems students are likely to encounter.

I have been following you for the past 25 years (or more). You are my favorite math expert. I know all of the work I did following you is what spurred my love of elementary math curriculum and encouraged me to seek certification in that field (and become an elementary math specialist myself). THANK YOU!!!!

Thanks for your comment, and for your feedback that made my day.