Tag Archives: measurement

Decimals before fractions? [TDI 1]

The Khan Academy knowledge map got me thinking about this recently, but the basic question at the heart of this Institute has been on my mind for a very long time.

Does it make sense to study decimals before fractions?

Decimals lie beneath addition and subtraction in the hierarchy. Fractions are not even in this part of the map; they are far off to the lower left.

The Khan Academy knowledge map. Decimals lie beneath addition and subtraction in the hierarchy. Fractions are not in this part of the map; they are far off to the lower left.

We do not have to answer that question right away. Indeed I do not think that there is a simple answer. I will argue in the coming weeks that the preponderance of theoretical and empirical evidence points to no.

You are not obligated to agree with me.

As I worked on formulating an argument the other night, I tried to make my question more concrete. Here is what I came up with (via Twitter):

Now, Twitter is a medium that makes nuance difficult.

So let’s strive to find nuance, subtlety and complexity in this conversation.

That last question is an important one for me. Traditionally, U.S. curriculum has had students working with decimals before they work seriously with fractions. Khan Academy isn’t going against the curricular flow in this area. What this means is that one-tenth is the first fraction students study. Is this justified?

The arguments in favor of studying decimals before fractions include these:

Place value. Decimals are the logical extension of the whole-number place value system. Just as you go from 1 to 10 to 100 by moving one place to the left, you also go from 100 to 10 to 1 by moving one place to the right. When you move left, the value of the place is multiplied by a factor of 10; when you move right, the value of the place is divided by a factor of 10. Decimals just continue that process.

Money. Children come to school with experiences involving money. They know what one dollar is; they know that 10 dimes make up a dollar; they have seen $1.25 and can talk about what that means. As a result, decimals are part of children’s everyday experience in a way that (say) sevenths are not.

Measurement. Metric measurements (and many but not all Imperial measurements) are expressed in units and tenths of units. Children are familiar with the meaning of “12.2 fluid ounces” or “3.2 meters”. So it makes sense to operate on tenths and hundredths even before formalizing the underlying mathematics of fractions.

How say you? Are these powerful arguments for you? Have I missed any arguments in favor of studying decimals before fractions? Do you have evidence to bring to bear on the question of whether it makes sense to study decimals first? Can you provide curricular examples to support (or refute) my claim that U.S. curriculum typically presents decimals before fractions? Can you provide an international perspective for us?

Follow this link to the announcement of this course for more information.

Instructions for joining the course:

This course has enabled open enrollment. Students can self-enroll in the course once you share with them this URL:  https://canvas.instructure.com/enroll/MY4YM3. Alternatively, they can sign up at https://canvas.instructure.com/register  and use the following join code: MY4YM3

Units, attributes and four-year olds

From mrdardy in the comments recently:

Slightly off topic, but I wanted to share a conversation with my soon to be four year old daughter from this past weekend, We were on a long car drive and she was asking how far we were from our hotel. I replied that we were twenty minutes away. Later in the pool she was jumping to me from the pool steps and commanding me to back up some. I asked her how far I should go and she told me to be five minutes away. I said “Do you mean five feet away?” and she replied, firmly, that she meant for me to be five minutes away. I am wrestling with whether I think this is just charming and (semi) clever on her part or whether I need to start answering her pleas in the car with distances. Curious to hear some ideas on this.

I am happy to weigh in here.

Anna Sfard describes knowledge as participation in a discourse and learning as changes in that participation. That is, we can measure whether someone knows something only to the extent that they can talk in ways that adhere to the norms of other knowledgeable people. And when these behaviors change to conform more closely to these norms, we can say that they are learning.

Nowhere is this more clearly demonstrated than in the learning of young children.

The four-year old in question here (let’s call her “Little Dardy”) is trying very hard to participate in conversations about measurement. Measurement, though, is a challenging and rich domain. 

mrdardy outlines two scenarios in which the concept of how far comes up for Little Dardy. It shouldn’t be at all surprising—considering Sfard’s model—that she answers a distance question in the same way her father had earlier on. She has taken his example in using units of time to discuss how far something is.

My approach would not be to avoid using units of time to answer the question how far? After all, people do this frequently; it is part of the discourse of measurement.

No, I would use this tension to encourage Little Dardy to think about the two attributes in question here: time and distance. It might go something like this…

Little Dardy: (four years old) Back up, Daddy!

Daddy: This far?

LD: More!

D: Here?

LD: More! You need to be five minutes away!

D: Do you mean five feet away?

LD: No! Five minutes!

D: OK. Tell me when I’m there. But then don’t jump right away; I want to ask you a question before you do. [Daddy backs up slowly...]

LD: OK! There!

D: Right. Here’s my question: Do you think it will take you five minutes to get to me from where you are?

LD: Yes.

D: Do you know how long five minutes is?

LD: That far.

D: No, no. Can you think of something we do together that takes five minutes?

LD: No.

D: It takes us about five minutes to read [INSERT TITLE OF FAVORITE PICTURE BOOK HERE] together. Do you think it will take that much time for you to get to me?

At this point, I have no idea how Little Dardy will respond (which is what fascinates me so much about talking math with kids). I do know that pretty soon, she is going to want to jump, and that whether that’s right away or after a few more exchanges doesn’t really matter.

What matters is that she’s been asked to think.

This line of discussion lays the foundation for thinking about distances, times and their relationships to each other. It supports Little Dardy’s attempts to participate in the discourse of measurement.

My recent conversation with Tabitha about the height of our hill was in a similar spirit; we worked on the meaning of height when she asked me to lie down on the hill.

How tall is the hill? [summer project]

Our house in St Paul sits on top of an odd hill; higher than others around it. Historical reasons for this are murky but it makes the place easy for guests to find. One of my least favorite tasks in all of my domestic life is mowing the hill.

20130724-133524.jpg

For a while now, the precise height of this hill has been the subject of family speculation. One recent lazy summer afternoon, Griffin (8 years old), Tabitha (6 years old) and I found ourselves hanging out on the hill with not much to do.

Me: How tall do you two think the hill is?

Tabitha (6 years old): Five feet.

Griffin (8 years old): I don’t know.

Me: How about this: Which do you think is taller; me or the hill?

T: The hill.

Me: Wait. I’m six feet tall. How can the hill be 5 feet tall AND taller than me?

G: You’re six feet, one inch.

Me: Right. Even so…

T: Oh. I don’t know how tall the hill is, but I think it’s taller than you.

Me: Why?

T: Lie down.

20130724-133733.jpg

T: See?

Me: Yeah, but just because it’s longer than me doesn’t mean it’s taller than me.

Tabitha seems puzzled by this distinction. Griffin is standing on the sidewalk at my feet.

Me: Look at Griffy’s eyes. Is he looking up or down at my eyes right now?

T: I can’t really tell.

I stand up, right next to Griffy, who cranes his neck back to look me in the eye.

Me: Now?

T: Ha!

I lie back down on the hill.

Me: So how come there’s a difference?

T: You’re lying down now, so that’s not really how tall you are.

Me: So how can we decide whether I am taller, or the hill is?

Nothing much occurs for the next minute or so. We are distracted by butterflies, the edible nature of clover flowers and other wonders of Minnesota’s too-short summers.

Me: Hey! Let’s try this. Tabitha, you go to the top of the hill.

She does, and she stands there, looking down on me with a self-satisfied smile on her face.

Me: OK. So you plus the hill are taller than I am. What about just the hill?

T: I don’t know.

Me: Lie down.

She does, although it takes a few tries to achieve the desired position by which she can look at me from roughly the level of the top of the hill.

Me: Are you looking up or down at me?

T: I can’t tell.

Griffin takes his turn at the top of the hill. He, too, is unsure.

Me: So how can we be sure?

T: You know, Daddy, I don’t really need to know this.

Me: You’re right. You don’t. Nor do I, really. But I have always been curious how tall the hill is. Aren’t you?

G: We could measure a step, then use the number of steps to figure out how tall it is.

I obtain a tape measure.

We determine that each step is 7 inches tall. We notice that the bottom step is shorter than the rest and measure it at 5 inches. Griffin laboriously counts the steps, finding that there are eight of them, plus the smaller one.

G: So what is that altogether?

Me: What? You can do this.

G: Do you know whether you are taller than the hill?

Me: Actually, yes I do, even though I don’t know exactly how tall the hill is.

G: If I figure it out, will tell me whether I’m right?

Me: Yes.

G: [Far too quickly for me to be convinced he has run any computations at all] OK. The hill is taller.

Me: How do you know?

G: Hey! You said you would tell me!

Me: That’s part of doing the math!

G: OK.

A long, thoughtful pause ensues.

G: Eight eights is 64, plus 5 is 69. So you are taller.

Me: But you need eight sevens, which is 56.

G: Oh. Right. Plus 5.

Me: Yes…?

G: Tell me.

Child, please.

Me: Seriously? You can do 56 plus 5.

G: 61.

Me: Yes, and I’m 73 inches tall.

Tabitha, despite her protestations about not needing to know, has been paying attention all along.

T: You’re taller than the hill?

Me: Yes. See? I told you it was interesting.

G: You knew you were taller?

Me: Yes. But I didn’t realize it was by a foot. I thought it would be only by a few inches.

G: How did you know?

Me: Because I look down—only slightly—but I look down at the top of the hill.

In a few days, we will return to the topic of the State Fair Giant Slide and see whether these techniques generalize in my children’s minds.

Summer project

The Minnesota State Fair is a fabulous event (Twelve days of fun ending Labor Day!). Rachel and I love the Fair, and we have passed this love along to our children.

Griffin must have been thinking about the wonders of the State Fair as summer slowly (oh, so slowly!) unfolded on our fair state. He asked a question at breakfast one recent morning.

Griffin (eight years old): How tall is the Giant Slide?

Me: Good question. I would guess…40 feet. What’s your guess?

G: 45 feet.

OK. That’s a mistake. We should have written our guesses down privately to avoid influencing each other. Oh well.

Me: Let’s look it up.

Google returns nothing useful. It does return this awesome video, though, which we watch together.

Me: I found lots of information mentioning the Giant Slide, but nothing on its height.

G: Measure it yourself, then!

Me: Good idea. How should we do that?

G: We’re gonna need a lot of tape measures put together.

This will be a summer project for us: Measuring stuff without putting a ruler next to it. I’ll report on our progress in this space.

Measurement, explored

This idea started with someone else, but I do not remember his name. I believe he’s a shop teacher in a Twin Cities suburb. Inver Grove Heights, maybe? In any case, he was in a professional development session I was helping to run this year on the topic of fractions. We had a conversation over lunch in which he recounted a lesson he did that became the basis of the activity I am about to describe. If I can dig up the originator, I’ll revise to give credit.

In any case, while the kernel of this idea originated with someone else, I have given it the usual OMT treatment—expanding and complexifying in many ways.

Regular readers will know that I am always in search of ways to get my future elementary teachers to explore old ideas in new ways. Consider the cases of place value and the hierarchy of quadrilaterals. In that spirit, I give you the measurement exploration extravaganza. Do with it what you will.

The premise

Groups of three are each given a dowel (or, in this year’s case, a paper strip). The dowels vary in length. The lengths are chosen to provide a useful combination of compatability and incompatability. One may be 9 inches long, while another is 15 inches long. Choose numbers according to the skill level and age of your students (and yourself!)

But-and this is important-THESE LENGTHS ARE NEVER SPOKEN OF! You will never refer to these dowels using standardized lengths.

Each group names its unit. In recent semesters, we have had:

  • Stick
  • Woody
  • Shroydelshnop
  • Oompa Loomp
  • BOG
  • Ablue
  • Pen
  • Et cetera

shroydleshnop

The members of the group measure some stuff with their units. They make a tape measure to use for this purpose, and they decide how long a tape measure they would like to have.

For example How tall are you in Sticks? requires (in all likelihood) a tape measure that is several Sticks long. Well, it does not require such a thing, but such a thing facilitates this measurement.

At this point, students are measuring only with their own units. It usually occurs to them to subdivide the unit in some way, and they will frequently report out fractions of (say) a Stick.

Next, each group is responsible for creating a partitioned unit from their original. They choose how many of these smaller units make up the original, and they name the smaller unit.

And then they create a composed unit from their original. Again, the choice is theirs to determine the number of original units that make up a composed unit. And again they are tasked with naming the composed unit.

interlude for important observations

The fun has only just begun and already we stumble upon some beautiful insights. Among them are these:

  1. Students nearly always partition in 4ths, 8ths and 16ths.
  2. Students almost never partition into 10ths.
  3. Students may group in threes or sixes, but they never ever partition this way.
  4. Students rarely think to group the same way they partition. That is, if they made 8ths, they might very well group in sixes. The convenience that would be afforded by consistency does not tend to occur to them in advance.

back to the instructional sequence

Now that we have the units, we need to measure some stuff. I typically choose things in our classroom environment. It is important that we all measure the same things and that these things range from smaller than the original unit to larger than the composed unit.

We need to express our measurements in (1) partitioned units only, (2) original units only, and (3) composed units only.

unitsThis semester I had students look at this table and I asked What do you notice? and What do you wonder? (These questions are, of course, not original to me. But this was a productive place to ask them.)

Working across systems

Next, it’s time to switch things up. We put the table away. Each group passes their  original unit, together with instructions for creating a partitioned unit and a composed unit (and the names of these) to another group.

Now each group is charged with these tasks:

  1. Get to know the three units that have been handed to you.
  2. Express relationships between your units and these new ones.
  3. For each thing you measured (table, licorice fish, etc.), make this prediction: If you were to measure that thing with these new units, would you end up with a greater or lesser value than when you measured in your own units? (In this step, do not compute; make a qualitative comparison instead.)
  4. Compute your height in these new units, and compute at least 6 of the measurements in the grid.

You have never seen such fraction computation work as proceeds from this sequence of tasks. 

Now we list these computed measurements on the board, compare to the table we generated earlier and discuss reasons for discrepancies.

We write about these reflection questions:

  1.  How do your three units compare to a standard measurement system?
  2. How is using someone else’s units like (or unlike) converting between standard and metric systems?
  3. How did your choices for partitioning, composing and naming support or impede your work?
  4. What do you need in order to be able to do these computations on your own?

On to area

Next, students build each of their units into square units.

We consider the essential questions:

  1. How many square partitioned units in a square original unit?
  2. How many square original units in a square composed unit?
  3. How many square partitioned units in a square composed unit?
  4. Most importantly: How do you know each of these?

Sample student observations at this point: 

  • Wow. The square partitioned unit looks a lot smaller relative to the square original unit than I expected.
  • Oh no! Why did we decide to put so many original units together to make the composed unit?

Now we measure something. 

This time around, I had them measure the area of a whiteboard in our classroom. Not the most exciting measurement to make, but straightforward and accessible. Working with these new square units is challenging enough; no need to get too fancy. It is important that the measurement be concrete and tangible, not abstract.

Students are encouraged to use known relationships in order to avoid tedious measurements, and to measure in order to avoid tedious computations.

Importantly (I think), most students want to use these square units to measure, rather than to measure with their tape measures and compute.

summary

We use these experiences to discuss differences—both practical and conceptual—among measuring by (1) iterating and counting units, (2) using tools, and (3) computation.

We reflect on what these experiences can tell us about working within and across measurement systems.

We build on our fraction work and on the meanings of multiplication and division that were the focus of the preceding course.

I have not had students move to cubic units.