So you have here a fairly straightforward carnival estimation game, which I decided to complicate by filling up a smaller container with the same kind of (horrid) malted eggs and making that quantity known.

I surveyed my students, my math-department colleagues, some of their students, my principal, and the central office staff. A little over 100 guesses all told. I tagged each guess with the following metadata:

- name,
- guess type (gut check, visual estimate, math computation),
- job description (student, math teacher, staff member, principal),
- current math class (eg. Algebra 1, Geometry, AP Calculus, etc.),
- grade level (freshman, sophomore, junior, senior),

I showed my students the raw data and asked them what they wanted to know. I wrote their questions on the board.

- who won?
- who guessed worst?
- what was the ranking of everyone in between?
- what type of people used
*math computation*for their guesses? - were there any tied guesses?
- what was the highest/lowest guess?
- which grade level guessed the most?
- which grade level guessed the best?

**Define “Bounty”**

I said I was offering a “bounty” for answers to those questions and asked them to define the term. Some kids had seen *Dog the Bounty Hunter* and explained it from that angle. I assigned each question a point value that corresponded roughly to a) the difficulty of the question and b) its relevance to my objective — **how are absolute value and percent error useful for calculating accuracy?** I offered 20 points for a *picture* of an interesting fact. (See “Interesting Pictures” below.)

They had to scrape together 100 points for the day and I offered extra credit for initiative, divergent thinking, etc.

**What Happened**

Students worked in pairs on laptops. They downloaded an Excel sheet with all this data, including the real name of every guesser. Naturally, they were into that.

The great part about a sample size of one hundred guesses is how easy it was to determine which groups were taking a tedious, manual approach to these questions and which were using Excel’s built-in capability for sorting and calculating. I circulated the classroom and could tell that a group was ready to learn more about Excel because they were using *hash marks* to count up every freshman, sophomore, junior, and senior. Those students were wandering the desert on foot, ready for the water, compass, and camels I could offer them.

Likewise, I saw another group of students subtracting all one hundred guesses from the actual answer (1831) *one at a time* on cell phones. It didn’t take much to convince them to experiment with another approach.

My favorite conversations with students centered around a definition of “accuracy,” as in, “who were the top ten most accurate guessers?” Our earlier trick of just subtracting the guesses from the answer messed with Excel’s sort mechanism, unhelpfully stacking positives on top of negatives, when, really, we didn’t care if you guessed 100 eggs too high or 100 eggs too low. For our purposes, those two people tied

Two students were so close to constructing that operation themselves I had to bite off my tongue to keep from spelling the whole thing out (“ABSOLUTE VALUE! SUBTRACT AND THEN TAKE THE ABSOLUTE VALUE!!”) and then the bell rang. We didn’t graph anything. We didn’t get to percent error. Half the groups got to absolute value.

Off moments like this, I have determined my constructivism multiplier to be *four*, which is to say it takes me four times longer to bring a student to conceptual understanding through conversation and questioning in a social situation the student helped create than it does to get up in front of the class and simply give it to them straight, no chaser, through direct instruction and a handout of questions I wrote.

What I find maddening about conversations with committed constructivists (cf. the conversation here) is the reflexive assumption that educators choose direct instruction because they’re either power-drunk or self-obsessed or because they lack faith, courage, or high expectations. I can’t, personally, wave so dismissively at the massive institutional impediments to student-constructed learning.

**Interesting Pictures**

*Percent Error By Grade Level*

*Percent Error By Guess Type*

It’s worth pointing out here that “Math Computation” isn’t the same thing as “Correct Math Computation.” The most accurate guessers verified their correct math computation with a visual estimate.

*Percent Error By Math Class*

*Percent Error By Job Description*

That last graph is what I meant at TEDx when I said that math gives your intuition a certain vocabulary. The math teachers have a more descriptive vocabulary for expressing their own intuitions than the students do. This is also a fair answer to the question, “when will I ever use math?” You might not. You can live without it. But it makes a lot of intuitive tasks a lot easier. And you should also understand the risk that you’ll one day be fleeced by or passed over for those who know how to speak with that vocabulary.

**The Creative Feedback Loop Of Teaching**

Where else can you get this? In all of the creative fields that have ever tempted me professionally — I’m talking about graphic design, screenwriting, and filmmaking — ideas often take months to generate and refine, years to produce, and, in many cases, you can’t *do* anything with the feedback except hope it’s good enough to get you your next job.

With teaching, you can get any old harebrained idea on Friday, challenge your students with it Monday morning, then adapt it for your afternoon class based on feedback from the morning. The feedback loop is fast enough to give you whiplash. It’s so much fun, this job, it seems impossible sometimes that anyone could ever walk away from classroom teaching.

**The Grand Prize**

Not those horrid malted eggs, that’s for sure.

## 18 Comments

## Jackie Ballarini

March 20, 2010 - 8:12 am -I agree that the “constructivism multiplier” is greater than one. I’m not sure I agree with the four. (I think it may be higher). My question for you is how do you know when (and at what level) a student has achieved the conceptual understanding?

When I have a discovery activity and I’m circulating the room talking to each group, I feel I have a much better understanding of what each student knows than when I give them a few examples and circulate to check progress. In the first case, I get an understanding of both what they understand and what they’re able to do. In the second case, I usually only get an understanding of what they’re able to reproduce.

## Newteach

March 20, 2010 - 8:55 am -In my situation, I feel as though everything I’ve learned about teaching has proven to be almost diametrically opposed to what has worked in the classroom. Now, right up front I’ll say that I have the “low class” in a classic urban, high poverty, low parent education level (as in, the parents can rarely do basic 6th or 7th grade math either) setting.

Doing more constructivist activities generally leads to off-task behavior, arguing, etc. It also leads to wildly crazy answers that do not meet my posted requirement “Always ask: Does it make sense?”

Then again, if I’m very direct instruction-y, they are happy to wait for it, wait for it, wait for it and just copy when someone answers rather than work at it. (“It’s too much, it’s too hard” is a reflexive response, used for any level or amount of work).

This summer one thing I’m going to be mulling over is the idea of visualizing — how do I develop that ability in middle school age kids? I know that one way is to keep having them draw pictures…but it seems like I need a variety of strategies for developing a way to see in your head if you are adding/multiplying vs. subtracting/dividing or if your answer is going to be bigger or smaller than the numbers in the problem or if you are making pieces or creating a whole…

And I *think* that using constructivist ideas to develop those abilities may be what I’m aiming at. Constructivism for developing some bits of number sense and math intuition and straight up (hidden from curriculum cops) direct instruction for getting certain basic facts and procedures to stick in heads.

Or something like that.

## Dean Shareski

March 20, 2010 - 9:58 am -I like those eggs. That last photo made me sad.

## curmudgeon

March 20, 2010 - 10:35 am -Horrid eggs !

No matter how bad somethings taste to me, there will always be one kid who will sit there the WHOLE period, pop pop pop pop pop.

Okay, STOP. You’ve eaten half the jar. (How many calories was that? — here’s the bag so you can judge if you’ve exceeded your 150 cal of sugar quota.)

## Dana Huff

March 20, 2010 - 1:34 pm -My first thought after reading this post is that I wish you’d been my math teacher.

## Kate Nowak

March 20, 2010 - 7:29 pm -Four sounds about right.

## Tom

March 21, 2010 - 4:07 am -Have you given any thought to doing some control comparisons for constructivist vs direct? I know it won’t be exact – but things like comparing how the different classes do immediately and then long term on assessments. It’d also be interesting to see how the students (and you) feel about the different lessons. Would it be worth 4x the time if changed student interest enough? Does any of that interest change long term opinions? That sort of stuff . . .

There’s some conversation about doing this our district with tech integration because of the 1:1. People want “proof” that giving students laptops matters. I can think of lots of issues with our attempt but you might be able to avoid at least some of them as an individual doing it on your own.

## Oldteach

March 21, 2010 - 7:04 am -I’ll have to agree with Newteach. It sounds like we are teaching in a similar situation, and I also find my students best behaved and nearly 100% on task when the activity does not involve thinking.

We can’t give up, though, you need to keep at it and train their thinking muscles.

## sam b

March 21, 2010 - 9:57 am -“I also find my students best behaved and nearly 100% on task when the activity does not involve thinking.”

agree. but the aim of your class is not to have an hour of good behaviour.

totally admit i do the same thing and will often give routine, immediate near mindless tasks at the start of a lesson to get them settled.

## Mister2pi

March 21, 2010 - 3:00 pm -Even if the constructivist multiplier is 4 or even 5 I think I would take that any day over mindless tasks. Unfortunately the first couple of years of my teaching career involved lots of lectures and very little student interaction except for the notes they were scribbling. They had a big fat notebook but they didn’t know how to make sense of what was in it.

Think of your most worthwhile learning experiences. For me, none of them involve notetaking or being lost in a sea of bullet points. They are learning [I]experiences[/I]…meaning I had a chance to interact with the concepts and form a concrete link to them.

For example, take this year’s Algebra 2 class. I had an engineer form Edwards Air Force Base come down and wow them with stories of the Global Hawk, cool pictures, and two-wheeled robots controlled by a TI-84 Plus (www.smallrobot.com). Each student was given a robot and meter stick, and through some probing questions, were asked to see if they could determine the speed of their robot. Even my smartest sophomores had trouble linking the activity with D=RT.

But, now if I need to ask them about D=RT, I can say “Do you remember the robots?” instead of, “Do you remember that slide that I put up with the cool white background…yeah that one…remember what the 2nd bullet said?” Which one would you more easily remember?

## Tony

March 21, 2010 - 3:40 pm -A few days ago I listened to a RadioLab podcast that included a great story about the power of the *group* in guessing unknown quantities (I think in that case it was the weight of a bull). So I can’t help but wonder: did the 100 guessers guess pretty well, *collectively*?

## David B. Cohen

March 22, 2010 - 10:37 am -Always a treat to see what you’re up to. Now, does the volume of horrid candy coated malted chocolate eggs have anything to do with how quickly the checkout line moves at the grocery store?

## ClimeGuy

April 1, 2010 - 7:44 am -Here’s an applet that time tests you on your guessing ability:

http://www.ninjakiwi.com/Games/Puzzle/Play/Guesstimation.html

Can you use this to help young kids learn to guess better? I’m not sure.

## mona schraer

June 8, 2010 - 7:52 pm -I love this – I have been doing an excel unit for years – the kids always love it and they get involved, but I always give them the data set – this turned the unit around – to have the students make the data set themselves – come up with their own questions – both with the candy and then the celebrity ages – I do believe on the candy one the staff became more caught up and interested in the results then the students – a kick to have the school involved in one classes project – we ended up posting a big poster with graphs and statistical results for the school to see.

Anyway thank you for sharing all your ideas – they are bringing a new refreshing element to my class and to how I think about teaching mathematics – your ideas in a very low income middle school are keeping me and my students going this last month of school.

## Tim Chadwick

September 24, 2012 - 8:51 pm -I loved the bounty idea. Just showed my year 10s (14-15 yr olds) this article about a UK supermarket’s price promise

http://www.dailymail.co.uk/news/article-2207905/Shoppers-8-600-Asda-spree-free-Loophole-price-guarantee-scheme.html

We came up with lots of questions (assuming that he made 7p per pack of ornage juice and that he only bought orange juice- it could go simultaneous if you add extra products)

An extra 10 points were awarded for any graphs and use of algebra per question

– How many orange juice cartons did he buy?

– How many did he buy per day?

– Write the computer program that could work out the cost of the voucher (and work out if you get a voucher!)

– How many swimming pools could he fill (we used basic 25m dimensions) and to how high?

– How long would it take to drink all the juice?

(extra points for graphs showing different drinking rates, inverse proportion was used when i said he could have as many friends as he liked helping him)

ANd more I can’t remember. Thanks for the idea, I’ll use it regularly now!