Rich Tasks — Saving Space for Student Thinking

Lots of time has passed since I last posted. I have been up to my eyeballs in new curriculum planning/envisioning (fourth grade is new for me), union negotiations (I’m the chief negotiator for our local), and in creating portfolio entries for my attempt to achieve National Board Certification. While there are many stories to tell about what has happened lately, a recent post about by mentor, ether-friend Vicki Vinton jarred loose a post that has been rattling around in my brain for awhile.

Vinton talks about how she’s benefited from her connection with math colleagues who talk about “rich tasks.” In upcoming posts, she will talk about how to apply the idea of rich tasks to reading, too. According to Vinton, rich tasks are those that “provide multiple entry points”, “invite creative and critical thinking”, “spotlight…both processes and product…(to help) students better see the connection between means and ends”, and “promote student ownership.” In other words, they are the kind of tasks that a teacher loves to create and witness.

Like Vicki, I’ve also benefited from being in touch with math thinkers who seek to understand what students are thinking, what their misconceptions are, where the limits of their knowledge and skill lie, and how they approach/attack a math task. 1 The richer the task, it seems, the greater the opportunity to discover the edges of student thinking. And, truthfully, teaching gets really fun when we are near those edges.

If there is any subject that can create a rich task, science is one! And that’s where we are right now.

Recently we finished taking the IA Assessments 2 To celebrate, we’re learning a lot of science. 3 Our task has been to design and build an air-powered skimmer. 4

First, the students formed design teams. I asked them to create a logo and a slogan. That was an interesting task in itself. We “closely read” some logos and slogans that we found on line, how they tried to transmit those meanings through graphics and short text.

This part of the task helped me see many things, in particular how students tried to manage their own uniqueness as part of a group, but also their awareness of an audience outside themselves. Some were better able to imagine that outside audience than others. These degrees and kinds other-awareness were interesting grist for the teacher thought-mill, and seemed so connected to the crucial skills of listening and questioning that go into learning. A rich task like this helped me see the learners better. Anything that helps me understand them better as people seems to help my teaching.

Then, the students were given the “task” to create a wind powered skimmer that would go at least 60cm, but as far as possible. They set to work creating their initial sail designs. I observed and asked questions about the rationale for their design work. What do you think will happen? Why did you put that there? Why did you make this shape? Questions of that sort.

The project used a typical engineering design protocol, which we have used over and over again.

 

The design process we used.

The design process we used.

After the students designed their sails, they tested them, recorded the information from the tests, and re-designed to solve the problems that they discovered in the testing stage. I continued asking questions, helped them solve some of the group process issues that inevitably arose, and pushed them to examine the principles behind their designs. Although the students often “designed” based on principles (they had a gut-level sense of cause and effect as far as their design went), articulating those principles in more general ways is new to them and one of my goals for this project.

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Along the way, to help them focus on what we could generalize from our work, we gathered together periodically to talk about what design principles we have discovered about what make sails work well, which the children could use to help them re-designed their sails. I introduced them to vocabulary like these: friction, friction force, sail frontal area, hull, bow, stern, mast, torque…

Finally, we had a competition to see whose sail design would make the skimmer go the farthest. Each design team then evaluated the results of the competition. We created short videos (ostensibly to “send” to the EarthToy company president, IM Green) describing the design as accurately as possible, the outcome of that design as evidenced by the competition, their thoughts on why this outcome happened (using scientific terminology), and what their next design idea would be and why they expect that idea to be an improvement over the one they entered in the contest.

This reflection was an interesting task in itself, and putting it in video form allowed the children a chance to reflect on their presentation for future work.

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I realized from this process that I am so much happier, as a teacher, working backwards from student thinking to strategy instruction than I am starting with delivering the strategy without deeply knowing the student thinking first. I really enjoy giving the students a big, unwieldy problem or task whether it be in math, in science, or in reading. Then I like to probe the children’s thinking as they complete the task. This kind of teaching is less efficient and messier, I know. 5

This kind of rich science task could be transferred to other areas by analogy. For example, just this last week, as we were revising drafts of persuasive essays, one of the students mentioned that revising was a lot like what we were doing in science with our skimmers. I asked for elaboration (love that word). His response: It’s where we design something and then change it to make it perform better. If we can internalize that idea, that revision is re-design for the purpose of better performance, then perhaps it will make our writing better, too? Maybe, too, it’s at the heart of growth mindsets and those wonderful “principles” of math practice that the CCSS-Math outlines.

  1. I have learned a lot from the work of Dan Meyer, especially his 3-Act Math, Christopher Danielson, and Joe Schwartz.
  2. This is the test formerly known as ITBS, which I’ve heard some of the wittier middle school students re-name as a statement without a linking verb: it bs.
  3. Teachers understand how testing crowds out the best stuff. The best stuff is often messy and takes time. During testing, time is of the essence and messiness doesn’t fit the schedule.
  4. I wrote and received a grant to purchase a couple A World in Motion kits last year from the Governor’s STEM Start-up funding stream. We are using, and modifying, these materials.
  5. Interestingly, several of the math thinkers I follow are aware of how their focus on rich tasks, uncovering student thinking, and multiple solutions has spawned a counter-movement, one that focuses on explicit instruction of the most efficient algorithms, and “efficient” transmission of new knowledge. In Canada, for instance. Obviously, I’m much more aligned with the “new math” crowd than the efficient transmission camp. Like my math mentors, I feel that good teacher questioning should be designed to help me, and the children, understand the principles behind our thinking, principles we use to build our next level of understanding. I am pessimistic, though, that without strong conceptual understandings even the most efficient of algorithms are all that efficient in the long run.

Our Pencils are Moose: A Look at Classroom “Environment”

Moose Saying
Creative Commons License Photo Credit: Doug Brown via Compfight

Last week we pretended our pencils were moose and our classroom a habitat. In the process, we learned some math and science.

You see, we’ve had problems with these cheap-o pencils from WalMart, the ones the kids buy at the Back to School sale at the beginning of the year. Those pencils break and are really difficult to sharpen. The experience of using them can be so frustrating that last week Trysten came to me and wondered whether the cheap pencils were really cheaper than the more expensive, but more durable pencils.

Seizing the moment, we decided to see if we could answer Trysten’s question.

Enlisting the help of my partner’s expertise in biology–Beth is a plant ecologist–we conducted a “mark and recapture”-type experiment. I’m reporting some of the preliminary results and, more importantly, thinking about some of the interesting possibilities that this simple question posed for how I can teach math and science, with possible links to social studies.

The Experimental Design

We imagined our classroom to be a habitat and we imagined that the two kinds of pencils — the more expensive MegaBrands and the cheaper Dixons — were two different populations of animals. We cleared our classroom environment of all other pencils. We “released” 30 new, marked pencils of each brand back into the environment. We took an oath to be good scientists and use the pencils as we normally would, to not favor one pencil over the other because we wanted our experiment to not be biased by our choices. The pencils we released would “roam” around our classroom habitat doing whatever good moose-pencils normally do: draw, write, doodle, and wait patiently to be sharpened.

Recapture!

After a week in the “wild” we “recaptured” our moose-pencils to see if any of them “died” over the week. We also measured the remaining pencils to the nearest 1/10 of a cm to see how “sick or healthy” they were. Would there be a difference in the two groups of pencils?

Measuring to 1/10 of a cm helped us become more solid with both fractions and decimals. The kids were really engaged in that kind of measuring and they not only got a sense of how scientists collect measurement data, but why they need to measure to a high degree of accuracy.

We had to think about how to create a data table and how to record accurate information. Note the delegation of tasks and the use of a white sheet of paper to make measurement more accurate!

We had to think about how to create a data table and how to record accurate information. Note the delegation of tasks and the use of a white sheet of paper to make measurement more accurate!

Students measure pencils carefully to see how "healthy" they are after a week roaming the wilds of our classroom.

Students measure pencils carefully to see how “healthy” they are after a week roaming the wilds of our classroom.

Data Analysis

After measuring, we noticed that the cheaper Dixon pencils seemed shorter as a group. I asked the children how we could compare the two brands to see for sure if one lasted longer than the other. We talked a lot about how to do this; could we compare the biggest only? Each pencil from smallest to largest? The smallest only?

I told them that scientists often have to compare groups with each other just as we are having to do, and suggested that we needed to not just compare individual pencils, but we would need to somehow compare an entire group of pencils with another entire group. How could we do that?

One student, Isaac, mentioned that maybe we could compare the average size of the pencils in each group (or the median). That way we would be comparing all the pencils of one brand (by calculating the mean or median) with all the other pencils of the other brand.

This was a central insight about elementary-level statistics, and considers a question that scientists have to think about all the time! Also, while we have been figuring medians and means for awhile, the children have never actually understood why these values are important. To me, this insight about why scientists need to think about means and medians was one of the best things to come from our experiment.

Here are two pictures of the children figuring out the median pencil size by re-arranging sticky notes with the pencil sizes on them. To do this, we had to think about how to represent the “dead” pencils, and decided that their “health” should be 0.0 cm. They also had to think about simple things like how to construct data tables, and how to incorporate new data into their evolving medians as new research groups added more sticky notes to their growing collection.

Note all the hands actively involved in rearranging the sticky notes. There was a lot of great conversation. Students in the back were beginning to notice patterns between the two data sets.

Note all the hands actively involved in rearranging the sticky notes. There were a lot of great conversations. Students in the back were beginning to notice patterns between the two data sets.

Even re-arranging the sticky notes required the children to think through an organized way to do this evolving task as new data was coming as research groups completed their measuring.

Even re-arranging the sticky notes required the children to think through an organized way to do this evolving task as new data was coming as research groups completed their measuring.

Our results showed average pencil size of 13.3 cm for the MegaBrands (1 “death”) and 10.2 cm for the Dixons (four “deaths.)

Next Steps

Next, we will graph our data. We still haven’t decided how to figure out whether the cheaper pencils are actually cheaper. As one child asked: What if the cheap pencils were only a penny, but broke very easily, and the others were $1,000 but lasted a long time. Maybe it would still be cheaper to buy the cheap pencils.

Perhaps we will have to figure out how to use our “mark and recapture” data to predict when all of pencils will “die.” We will also need to gather information about price differences between the two pencils. Then we’ll probably need to compare our “death rates” with the difference in price. This will be difficult for the kids to imagine as they haven’t had experience with graphs as “predictors” of outcomes before, but I think the fact that they have now “lived” the experience of using the pencils, and have seen the differences in lengths that resulted from those days in the wild, that when we graph the data they will see how graphs render those differences visually. Perhaps then the children may be ready to draw a line on a graph down to zero and understand how a graph might help us understand a future event.

Finally, if the kids are interested, I’d love to turn this experiment into a look at how pencils are made and at the way consumer goods (and consumer choices) are influenced by price, which is influenced by the sourcing of the pencil wood and lead material, and probably also the labor costs. Where do the leads and woods for the cheaper Dixon pencils come from? This could be a fascinating way to look at global trade, the sourcing of raw materials, and the effect of labor cost on the manufacture of consumer goods.

 

 

Uncovering Misconceptions in Math: More on Decimals

In a recent post I talked about how difficult decimals are to learn. I asked the kids to explore the place value and fraction qualities of decimals by dividing a meter into 1/10, 1/100, and 1/1,000.

Well, I checked back to see how well those ideas were understood. I’m glad that I did because those conversations revealed an interesting way that a small number (5 of 21 in my class) had “encoded” the learning that we did that day.

Here’s the problem that I gave the kids: Pablo ran 9/10 of a mile in gym class. Walter ran 0.92 miles. Who ran farther. Explain your reasoning.

Most of the children identified Walter’s distance as longer, explaining that they were close, but that Walter ran 2/100 (0.02) of a mile farther. This showed some good grasp of the notion of place value in decimals. Here are a couple examples of that line of thinking.

A clear explanation of the difference between the two distances.

A clear explanation of the difference between the two distances.

Another fairly clear explanation of the difference in distances.

Another fairly clear explanation of the difference in distances. This one doesn’t identify the difference as clearly as the above example, but still shows good understanding of decimal place value.

However, several students appeared to have internalized the concept that the farther to the right from the decimal a numeral is, the smaller the piece was. So, for example, they had been impressed by the “smallness” of the 1/100 compared to the 1/10, the smallness of the 1/1,000 compared to all the other parts of a meter we looked at.

But, the students hadn’t deeply understood the idea of what that means in terms of place value, so they had formed a fuzzy understanding that any number that has numerals farther to the right must be smaller because the pieces are smaller.

In addition to this fuzzy thinking about relative piece size, their explanations contained a general sense that decimals were surprising, so the “normal rules” did not apply  to the right of the decimal point. In fact, by listening to their explanations, decimals were almost mirror images of “normal numbers,” a term this group used often in conversation. In this world, numbers that appear large (like 1/1,000) actually represent very small pieces. It is only one small step to assume that all numbers that appear larger are actually smaller.

Here are some examples of that kind of thinking. Notice they overlook the central idea of place value and substitute the misconception that decimals don’t follow the normal rules, so large numbers are actually smaller than small numbers.

Notice the discussion of the size of pieces, without incorporating place value concepts.

Notice the discussion of the size of pieces, without incorporating place value concepts.

Again, this explanation highlighted the idea of piece size, rather than place value concepts.

Again, this explanation highlighted the idea of piece size, rather than place value concepts.

All of this shows that while getting answers right or wrong is important in math, understanding the origins of student misconceptions and the function they serve for the learner’s developing understanding is crucial for the teacher’s next steps.

What we did on Thursday and Friday was explore the concept of place value, drawing and modeling what 92/100 vs. 9/10 (90/100) looked like, while still maintaining the notion that 1/100 of anything results in smaller pieces than 1/10.

 

Traces of Learning: Place Value and Decimals

Raven's Canvas
Photo Credit: Mike Beauregard via Compfight

Here’s the barest trace of learning for both the kids and me, like animals tracks on the snow.

How to teach decimals to fourth graders?

Decimals are both fractions and part of the normal place value system. Yet, I’ve found that decimals don’t fit neatly into either category. For example, if taught as a fraction of a whole (which they are), decimals require that kids deeply understand the idea of division of a UNIT (1) into 10 equal parts (for 10ths), and that smaller part into 10 equal parts (for 100ths), and on and on into ever smaller chunks of a whole unit.

The only real experience they’ve had with 10s is in school as the concept of place value, which is most often learned as a grouping concept not a division concept. Ten is a group of ten ones, 100 is a group of 10, tens (or 100 ones), etc.

Kids’ lived experience with division of objects is as objects divided into 1/2s, not 10ths. Cookies, pies, candy bars all get divided in half (or 1/2 again to make 1/4ths). The kids have a hard time transferring what they know about lived-fractions to  seeing 10ths as fractions.

More problems with teaching decimals.

Even though decimals are an extension of the concept of place value, they don’t fit very well there, either. Sure, each place that you go to the left on a number is 10x more than the place to the right, and that includes the decimal region of a number, too!

When I taught the connections between place value and decimals this week, I tried to emphasize this connection to place value by having the kids write series of numbers that contained decimals: 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, etc. The kids got it that this is just like any other kind of counting, but there is a decimal point added into the number. Each numeral to the left is 10 times the numeral to the right, and it doesn’t matter that a decimal point intervenes.

But when you get to the decimal side of the number, things get a little confusing for kids, if they think about place value. For example, when we count whole numbers everything is said as the total number of ONES, e.g. 345 is three hundred forty-five ONES. Kids learn this logic when they learn the number system.

However, when you start adding decimals, the UNIT itself appears to change. For example, the number 23.4 (23 and 4 tenths) has TENTHS as the last unit said. The number 23.45 (23 and 45 hundredths) has HUNDREDTHS as the last unit said. To say a number with a decimal component requires the speaker to alter the final word of the number (Tenths? Hundredths? Thousandths?) based on the place value of the decimal. That’s very confusing for kids.

So, what to do?

This week I started talking about decimals differently and I am trying to get the kids to say decimals differently, too. Instead of saying 23.45 as twenty-three and forty-five hundredths (which de-emphasizes the single UNITARY focus of a number), I’m trying to adopt a way of talking that re-emphasizes the UNITARY focus of numbers; I’m trying to say this: “twenty-three full (whatever the units are) and forty-five hundredths of one more.” My hope is that this language will help kids see that the number still revolves around declaring the amount of a single unit, except that the decimal side allows us to talk about units that aren’t complete.

Also, I thought the kids needed to get more experience tying the concept of division to the idea of place value so they could see that place value is more than just assembling groups of 10, but can also be dividing groups into 10ths.

So, we cut a paper tape into meter lengths. I asked the children to create other pieces of tape to lengths that were 1/10th, 1/100th, and 1/1000th of a meter. The goal was to give the kids experience with division into 10ths (to increase their lived experience of 10ths division, if you will) and also to help the kids get a visual, emotional sense of how the larger the denominator, the smaller the piece.

We’ll need to do more of this kind of division activity to help the children get a lived experience of division into 10ths, and how quickly small this process brings us.

Dividing a meter into 10ths, 100ths, and 1,000ths. Things get small, quickly!

Dividing a meter into 10ths, 100ths, and 1,000ths. Things get small, quickly!

1 meter 0.1 m , 1/10 m 0.01m, 1/100 m 0.001m, 1/1,000 m

1 meter
0.1 m , 1/10 m
0.01m, 1/100 m
0.001m, 1/1,000 m

 

Learning Math, Math and Learning

Deep Down Inside, We All Love Math T-Shirt Design
Creative Commons License Photo Credit: _Untitled-1 via Compfight

For some time I’ve been thinking about how to get more real-world math in our classroom. Our school district uses the Everyday Math program and I’ve been sometimes surprised at how little “everyday” there is to go along with the “math.”

We have participated in Franki Sibberson’s Solve It Your Way challenge, which the kids have really enjoyed. This month asked the question: Who in your family jumps the highest? We decided to use this as a whole class activity AND also as a way to practice some data collection and analysis skills we are learning in math class.

We moved through a scientific processes assuming Franki’s question as our own. We created a hypothesis, and thought a bit about how we would conduct the experiment so we could develop a list of materials to gather.  That’s where the thinking got interesting. We generated a list of materials — yardstick, jumpers, flat space, data table — but we had a difficult time thinking about how we could collect accurate data since a jump happens so quickly.

I had an idea that we might be able to use a video camera to slow the jump speed down, but I didn’t want to come right out and tell the kids that; I wanted them to grapple with the dilemma for awhile.

We talked about several ways we could collect data, including having someone lie on the floor and try to figure out how far up the yardstick children could jump. (That seemed hard to do since jumps happened so quickly.) We thought about extending our arms up a wall, measuring that extension, then jumping with something to put a mark against the wall. We’d subtract the jump mark from the non-jump mark. (I thought that was a pretty elegant solution to the problem since it wouldn’t rely on fast looking but on using mathematical differences. We could accurately measure the distances, too!) The problem with that solution, though, was we didn’t think we’d do our best jumping while trying also to make marks on a wall, so this method wouldn’t be an accurate measurement of our jumping abilities.

Next, I mentioned that we had some technology we could add to the mix, for instance we could add digital photography since I had both kinds of cameras in the room, and would that make a difference in how we designed the experiment?

The kids grappled with the implications of this new information. (“Maybe we could try to take a picture of the highest point of the jump?” But from their experience taking digital still photos, others said that wouldn’t be an easy thing to do.) Finally, one boy thought we might be able to try slow motion video. Others agreed. And that’s how we got our idea to create a video of our jumping.

When we looked back on the video we made, it was too hard to see the inch marks on the ruler so we re-jumped with inch marks clearly marked on a roll of tape paper. (An example of how scientists sometimes learn from our “failures.”) That change helped quite a bit! We were able to play the slow motion video until each jumper was at the top of the jump, stop the motion, and suspend the jumper in mid-air. (By the way, this was hilarious, seeing us all suspended in the air like that!)

What I learned from this — and what some of the children learned as well — was how closely connected the need for accurate data collection is to the design of the experiment. If flexible thinking is one goal of our learning together, then this was a good example of how we could achieve that outcome. We had to think how our materials and our need to slow down the act of jumping, which influenced how we did our experiment.

And this backward/forward type thinking reminded me of the work that we are trying to do in literacy as well, to be aware of the whole, rather than just the parts. There is no single, straight path to a solution, rather we hack our way through the jungle.

So, finally, here’s a short (3:22) video of our attempt to answer the question.

Wasn’t this kind of evil?

You wouldn’t know it by reading this blog, but besides reading and writing, I also like math and the content areas (social studies and science.)

Was it evil to give some kids this problem and ask them to find the median and the mean (average) without offering them anything but an explanation that average meant that all eight of the players would have the same number of home runs? 🙂

Math problem

Here’s what they did. You can see they started shaving off the high numbers and raising up the low numbers.

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It took them awhile to do it. Was it time well spent?

After they figured out the average home runs were 38 for the eight players, I showed them how they could do the same problem on their calculators. One of the kids said, “Hey, we were just dividing up the home runs equally! That’s division!”

I could have told them that they were doing division, but maybe discovering what an average means is better than figuring out how to arrive at one on a calculator?

Next steps are to have them use the averages to compare data sets. Math is fun.