Month: August 2013

Total 15 Posts

[Makeover] Boat Race

The Task


This is another task from MathWorks 10.

What Dave Major And I Did

I don’t have any huge beef with this task. I like that students get to pick their own route. Those kind of self-determined moments are tough to come by in math class. Here, the buoys are pre-determined but students get to make their own path around them. So we get the motivation that comes with self-determination but feedback isn’t the chore it would be if students got to choose the placement of the buoys also.

Establish a need for the bearing format. We’re going to take a cue from the research of Harel, et al. Rather than just introducing the bearing format as the next new thing we’re doing in math class, we’ll put students in a position to see why it’s necessary.

Offer an incentive for more practice. We’re going to make it really easy and enticing for students to try different routes, learning more about degree measure and bearings with each new route they try.

Raise the ceiling on the task. Rather than moving along to another context and another question, let’s stay right here in this one and do more.

Show this image.


Ask students to write down some instructions that tell the boat’s blind skipper how to navigate around the buoys and return to its original position. Don’t let this go on all that long. Whenever we’d like students to learn new vocabulary or notation, it’s useful for them to experience what it’s like to communicate without that vocabulary and notation, if only briefly.

Write the notation “50 miles at 60° South of East” on the board and ask them what they think it means. After some brief theorizing, send them to this website where they can test out their theories.

Then they can create a series of bearings that carry them around the buoys.


We’ve timed the boat’s path. But this isn’t the kind of timing you find on timed multiplication worksheets that freaks kids out for no discernible benefit. The timer here gives students feedback on their routes. The feedback is also easy to remediate and change. Feel free to try again and do better than your previous time. Or, if you’re feeling competitive, perhaps you want to try for the best time in class. (Or the worst time. That isn’t simple.)

Move on to the next page where we give you a series of bearings and ask you where the boat will come to rest. I find it tough to get inside 10 miles worth of error here.

If we wanted to draw this out even further, we might have:

  • featured multiple courses.
  • let students create their own courses and challenge their classmates.

What You Did

  • Frédéric Ouellet animated the boat in Geogebra. As with the work of a lot of expert Geogebraists, it seems as though the interesting mathematics is in making the animations or the sliders and has been done by the teacher, not the student.
  • L Hodge offers another Geogebra applet, one that puts more of the math onto the student.
  • Lindsay also asks her students to describe the path of the boat without yet knowing the vocabulary.

2013 Aug 21. It strikes me that some useful questions for provoking an understanding of degree measure would include:

  • What do you think “-20° North of East” means? Is there another way to write it?
  • What do you think “120° South of West” means? Is there another way to write it?

[Mailbag] Direct Instruction V. Inquiry Learning, Round Eleventy Million

Let me highlight another conversation from the comments, this time between Kevin Hall, Don Byrd, and myself, on the merits of direct instruction, worked examples, inquiry learning, and some blend of the three.

Some biography: Kevin Hall is a teacher as well as a student of cognitive psychology research. His questions and criticisms around here tend to tug me in a useful direction, away from the motivational factors that usually obsess me and closer towards cognitive concerns. The fact that both he and Don Byrd have some experience in the classroom keep them from the worst excesses of cognitive science, which is to see cognition as completely divorced from motivation and the classroom as different only by degrees from a research laboratory.

Kevin Hall:

While people tend to debate which is better, inquiry learning or direct instruction, the research says sometimes it’s one and sometimes the other. A recent meta study found that inquiry is on average better, but only when “enhanced” to provide students with assistance [1]. Worked examples actually can be one such form if assistance (e.g., showing examples and prompting students for explanations of why each step was taken).

One difficulty with just discussing this topics that people tend to disagree about what constitutes inquiry-based learning. I heard David Klahr, a main researcher in this field, speak at a conference once, and he said lots of people considered his “direct instruction” conditions to be inquiry. He wished he had just labelled his conditions as Condition 1, 2, and 3 because it would have avoided lots of controversy.

Here’s where Cognitive Load Theory comes in: effectiveness with inquiry (minimal guidance) depends in the net impact of at least 3 competing factors: (a) motivation, (b) the generation effect, and (c) working memory limitations. Regarding (a), Dan often makes the good point that if teachers use worked examples in a boring way, learning will be poor even if students cognitive needs are being met very well.

The generation effect says that you remember better the facts, names, rules, etc that you are asked to come up with on your own. It can be very difficult to control for this effect in a study, mainly because its always possible that if you let students come up with their own explanations in one group while providing explanations to a control group, the groups will be exposed to different explanations, and then you’re testing the quality of the explanations and not the generation effect itself. However, a pretty brilliant (in my opinion) study controlled for this and verified the effect [2]. We need more studies to confirm. Here is a really portent paragraph from the second page of the paper: “Because examples are often addressed in Cognitive Load Theory (Paas, Renkl, & Sweller, 2003), it is worth a moment to discuss the theory’s predictions. The theory defines three types of cognitive load: intrinsic cognitive load is due to the content itself; extraneous cognitive load is due to the instruction and harms learning; germane cognitive load is due to the instruction and helps learning. Renkl and Atkinson (2003) note that self-explaining increases measurable cognitive load and also increases learning, so it must be a source of germane cognitive load. This is consistent with both of our hypotheses. The Coverage hypothesis suggests that the students are attending to more content, and this extra content increases both load and learning. The Generation hypothesis suggests that load and learning are higher when generating content than when comprehending it. In short, Cognitive Load Theory is consistent with both hypotheses and does not help us discriminate between them.”

Factor (c) is working memory load. The main idea is found in this quote from the Sweller paper Dan linked to above, Why Minimal Instruction During Instruction Does Not Work [3]: “Inquiry-based instruction requires the learner to search a problem space for problem-relevant information. All problem-based searching makes heavy demands on working memory. Furthermore, that working memory load does not contribute to the accumulation of knowledge in long-term memory because while working memory is being used to search for problem solutions, it is not available and cannot be used to learn.” The key here is that when your working memory is being used to figure something out, it’s not actually being used to to learn it. Even after figuring it out, the student may not be quite sure what they figured out and may not be able to repeat it.

Does this mean asking students to figure stuff out for themselves is a bad idea? No. But it does mean you have to pay attention to working memory limitations by giving students lots of drill practice applying a concept right after they discover it. If you don’t give the drill practice after inquiry, students do worse than if you just provided direct instruction. If you do provide the drill practice, they do better than with direct instruction. This is not a firmly-established result in the literature, but it’s what the data seems to show right now. I’ve linked below to a classroom study [4] and a really rigorously-controlled lab study study [5] showing this. They’re both pretty fascinating reads… though the “methods” section of [5] can be a little tedious, the first and last parts are pretty cool. The title of [5] sums it up: “Practice Enables Successful Learning Under Minimal Guidance.” The draft version of that paper was actually subtitled “Drill and kill makes discovery learning a success”!

As I mentioned in the other thread Dan linked to, worked examples have been shown in year-long classroom studies to speed up student learning dramatically. See the section called “Recent Research on Worked Examples in Tutored Problem Solving” in [6]. This result is not provisional, but is one of the best-established results in the learning sciences.

So, in summary, the answer to whether to use inquiry learning is not “yes” or “no”, and people shouldn’t divide into camps based on ideology. Still unanswered question is the question when to be “less helpful” as Dan’s motto says and when to be more helpful.

One of the best researchers in the area is Ken Koedinger, who calls this the Assistance Dilemma and discusses it in this article [7]. His synthesis of his and others’ work on the question seems to say that more complex concepts benefit from inquiry-type methods, but simple rules and skills are better learned from direct instruction [8]. See especially the chart on p. 780 of [8]. There may also be an expertise reversal effect in which support that benefits novice learners of a skill actually ends up being detrimental for students with greater proficiency in that skill.

Okay, before I go, one caveat: I’m just a math teacher in Northern Virginia, so while I follow this literature avidly, I’m not as expert as an actual scientist in this field. Perhaps we could invite some real experts to chime in?

Dan Meyer:

Thanks a mil, Kevin. While we’re digesting this, if you get a free second, I’d appreciate hearing how your understanding of this CLT research informs your teaching.

Kevin Hall:

The short version is that CLT research has made me faster in teaching skills, because cognitive principles like worked examples, spacing, and the testing effect do work. For a summary of the principles, see this link.

But it’s also made me persistent in trying 3-Acts and other creative methods, because it gives me more levers to adjust if students seem engaged but the learning doesn’t seem to “stick”.

Here’s a depressing example from my own classroom:

Two years ago I was videotaping my lessons for my masters thesis on Accountable Talk, a discourse technique. I needed to kick off the topic of inverse functions, and I thought I had a good plan. I wrote down the formula A = s^2 for the area of a square and asked students what the “inverse” of that might mean (just intuitively, before we had actually defined what an inverse function is). Student opinions converged on the S = SqRt(A). I had a few students summarize and paraphrase, making sure they specifically hit on the concept of switching input and output, and everyone seemed to be on board. We even did an analogous problem on whiteboards, which most students got correct. Then I switched the representations and drew the point (2, 4) point on a coordinate plane. I said, “This is a function. What would its inverse be?” I expected it to be easy, but it was surprisingly difficult. Most students thought it would be (-2, -4) or (2, -4), because inverse meant ‘opposite’. Eventually a student, James (not his real name), explained that it would be (4, 2) because that represents switching inputs and outputs. Eventually everyone agreed. Multiple students paraphrased and summarized, and I thought things were good.

Class ended, but I felt good. The next class, I put up an similar problem to restart the conversation. If a function is given by the point (3, 7), what’s the inverse of that function? Dead silence for a while. Then one student (the top student in the class) piped up: “I don’t remember the answer, but I remember that this is where James ‘schooled’ us last class.” Watching the video of that as I wrote up my thesis was pretty tough.

But at least I had something to fall back on. I decided it was a case of too much cognitive load–they were processing the first discussion as we were having it, but they didn’t have the additional working memory needed to consolidate it. If I had attended to cognitive needs better, the question about (2, 4) would have been easier, and I should NOT have switched representations from equations to points until it seemed like the switch would be a piece of cake.

I also think knowing the CLT research has made me realize how much more work I need to do to spiral in my classroom.

Then in another thread on adaptive math programs:

Kevin Hall:

My intention was to respond to your critique that a computer can’t figure out what mistake you’re making, because it only checks your final answer. Programs with inner-loop adaptivity do, in fact, check each step of your work. Before too long, I they might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I can’t even sit with each student for 5 min per day.

Don Byrd:

I have only a modest amount of experience as a math teacher; I lasted less than two years — less than one year, if you exclude student teaching — before scurrying back to academic informatics/software research. But I scurried back with a deep interest in math education, and my academic work has always been close to the boundary between engineering and cognitive science. Anyway, I think Kevin H. is way too optimistic about the promise of computer-based individualized instruction. He says “It seems to me that if IBM can make Watson win Jeopardy, then effective personalization is also possible.” Possible, yes, but as Dan says, the computer “struggles to capture conceptual nuance.” Success at Jeopardy simply requires coming up with a series of facts; that’s highly data based and procedural. The distance from winning Jeopardy to “capturing conceptual nuance” is much, much greater than the distance from adding 2 and 2 to winning Jeopardy.

Kevin also says that “before too long, [programs with inner-loop adaptivity] might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I can’t even sit with each student for 5 min per day.” I’d say it’s likely programs might be better than teachers at that “before too long” only if you think of “identifying a mistake” as telling Joanie that in _this_ step, she didn’t convert a decimal to a fraction correctly. It’ll be a very long time before a computer will be able to say why she made that mistake, and thereby help her correct her thinking.

2013 Aug 14. Christian Bokhove passes along an interesting link summarizing criticisms of CLT.

[Makeover] Shipping Routes

As the summer winds down and #MakeoverMonday comes to an end, we’re going to crank up the difficulty around here. For the final three makeovers, I’ve commissioned work from some of the best people I know working in math, education, and technology: Dave Major, Evan Weinberg, and the team from Desmos.

The Task


This task is from McDougal Littell Middle School Math Course 1, 2005. [via Chris Robinson]

What Dave Major & I Did

TLDR: Here’s the 101questions page.

Lower the floor. The task currently jumps straight to the question of calculation. We should head in that direction but start with other interesting, easier questions also.

Enable pattern-matching. I could tell students what to look for here and how to approach the problem. I could show a few worked examples. For example, ones where:

  • one boat’s time is a factor of the other. (eg. 2 seconds and 4 seconds.)
  • the boats’ times are coprime. (eg. 3 seconds and 11 seconds.)
  • the boats’ times have a common factor. (eg. 6 seconds and 10 seconds.)

Two problems there:

  1. Some students will need more than just three examples to determine a pattern.
  2. My selection of those particular examples – that is, my decomposition of the entire solution space into just three categories – did a lot of the intellectual heavy lifting for my students. They need to decide on those three categories and come up with a rule that takes them all into account.

That isn’t to say I’d just “let them figure it out.” If a student just tries the first example and says, “It’s easy. It’s always the longer of the two times.” I can then say, “Great. But try that on several more examples and make sure it works.” (It won’t.) Or I can suggest one of the other two categories. But I’d rather not offer those categories before the student has even considered why she might need them, or even the fact that there are different categories.

Raise the ceiling. Our textbooks need fewer tasks and they need deeper tasks. The second fix would enable the first. Rather than jumping to another arbitrary context for another arbitrary example of cofactors, let’s stay in this context and extend it, developing the concept more for students who are ready for it.

Prove math works. It’s one thing to solve the original task for 150 seconds and find that answer in the back of the book. It’s another thing to watch the answer play out in front of you.

I’d ask students to watch this video.

I’d ask them, first, if they thought the boats would ever return to shore at the same time. The task gives that answer away but, me, I’d rather get every possible conception on the table so long as it doesn’t cost me too much time. “If you think they’ll return at the same time,” I’d then ask them, “write down how long you think it’ll take to see that happen.”

Then I’d send them over to Dave Major’s Shipping Route Simulator™. “Make up some boat time examples for yourself. Watch what happens. Make a table. Tables are useful for organizing data like this.” (We’ve intentionally set up the simulator so the domain maxes out at 10 minutes.) Then I’d tell them to pick two boat times and try to figure out what the answer will be before they check it by running the simulator. Whenever they’re ready, I’d ask them to tell me how long it’ll take the original boats to return to shore together and how they know.

I’d ask students who finished quickly:

  • Could you create two boat times so that the boats would never return to shore at the same time? Prove it. (Incidentally, this is one way I try to “be less helpful” – an expression that drives a certain set of math educators and mathematicians up the wall. Why give away the fact that the boats have to meet again? That’s an interesting question. Don’t be so helpful.)
  • What if you had three boats? Four?
  • What if the boats didn’t have whole number shipping times? What if one boat made its route in 2.5 minutes and another boat made it in 8 minutes?

Then I’d show the answer:

What You Did

In the preview post, most commenters seemed content to add elements to the word problem itself – adding a sentence about refueling schedules for motivation or turning the whole thing into a debate between two people about whether or not the boats will both return within the hour.

I’m sure that’ll have some effect on motivation and cognition but I’m not sure how large of an effect it’ll have or in which direction.

William Carey took a different approach:

I wonder whether a video of two bouncing balls or two oscillating springs or two swinging metronome hands would capture the idea of factoring to figure out when two cyclic phenomena will be in sync? That seems like it’s the perplexing bit of the problem.

Jim Pardun:

It reminds me of sitting in the left hand turn lane trying to figure out how often the turn signals will match up on the cars.

Not for nothing, I gave Jim’s example a shot some time ago and abandoned it. With most cars, the frequencies are so close that they converge again rather quickly. Part of the appeal with the ships is that it takes a really long time for them to converge.

If you’d like to see what goes into my rubbish bin, here’s what would have been “Turn Signals,” with two cars, three cars, and eleven cars.

2013 Aug 13. I don’t say this enough, but students should walk away from this lesson with a definition of “coprime” and “cofactor” written in their notes and, more ideally, stuck in their heads. Those definitions should come in the debrief of this conceptualizing activity, though, not in its introduction.

Great Classroom Action


I’m still clearing some links out of the filter, trying to get fresh for the new school year. Some great classroom action from the last school year.

Amy Zimmer shows us how to take a repetitive exercise worksheet and wring some more cognitive demand out of it:

My students were looking for and making use of structure, my students were constructing viable arguments and critiquing the reasoning of others, my students were looking for and expression regularity in repeated reasoning! I know just how geeky this sounds, but man, it was beautiful!

More adventures from Prof. Triangle Man in measurement in the elementary grades:

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 compatibility and incompatibility. One may be 9 inches long, while another is 15 inches long. But-and this is important-these lengths are never spoken of! You will never refer to these dowels using standardized lengths.

Bowman Dickson helps me see the benefit of starting at a low rung on the ladder of abstraction, even in highly abstract contexts like calculus:

So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning.

Evan Weinberg used cell phones and TVs to drive calculations of similar figures.