## “I’m gonna use my formula sheets and that’s the only way I’m gonna do stuff.”

The New York Times looks at the dismal testimony of an “accident reconstructionist”:

The “expert witness” in this case would not answer questions without his “formula sheets,” which were computer models used to reconstruct accidents. When asked to back up his work with basic calculations, he deflected, repeatedly derailing the proceedings.

Watch the video. It’s well worth your time and I promise you’ll see it in somebody’s professional development or conference session soon. It offers so much to so many.

And then help us all understand what went wrong here. What’s your theory? Does your theory explain this catastrophe? Does it recommend a course of action? If you could go back in time and drop down next to this expert as he was learning how to make and analyze scale drawings, how would you intervene?

BTW. Can anyone help us understand how the expert came to the incorrect answer of 68 feet?

BTW. Hot fire:

The motorcyclist’s lawyer filed a counter-motion to refuse payment to the expert witness. It contained the math standards for Wichita middle schools.

[via Christopher D. Long]

2016 Jan 2. The post hit the top of Hacker News overnight.

2016 Jan 2. One of the Hacker News commenters notes that the actual deposition video is available on YouTube.

gasstationwithoutpumps offers one explanation of the error:

3 3/8″ at a 240:1 scale gives 67.5′ which rounds to 68′

It is easy to mix up 3/8″ and 3/16″, which is one reason I prefer doing measurements in metric units.

katenerdypoo offers another:

It’s quite possible he accidentally keyed in 6/16, which when multiplied by 20 gives 7.5, therefore giving 68 feet. This is also a reasonable error, since the 6 is directly above the 3 on the calculator.

Jo illustrates a fourth grader’s process of solving the scale problem.

Robert Kaplinsky chalks this up to pride:

Lastly, it’s worth noting that eventually the heated conversation shifts from the actual math to whether or not he will do it or can do it. At that point it seems to become a pride issue.

Alex blames those awful office calculators:

The reconstructionist is given an office calculator, which doesn’t even have brackets. He needs to enter a counter-intuitive sequence of “3/16+3” to even get the starting point. When I was at school I remember being aware that most people wouldn’t be able to handle that kind of mental contortion. They’d never been asked to.
So what’s the problem, and how might we solve it? Well, the man’s been given the wrong tool for the job. He’s never been asked to use the wrong tool before & so this throws him. This makes him defensive and he latches onto an excuse about formula sheets.

The motorcyclist’s lawyer is the unrelenting classroom didactic whose motivation is based on making his student look and feel stupid. I was waiting for Act 2 where the lawyer would jump up, grab his felt marker, and demonstrate just how easy he can show the procedure.

Anna:

Interesting note: my grade 7 math class is in the middle of our unit on fractions, decimals, and percents, so I showed them this video so we could work on the problem. I thought they’d get a chuckle out of it and feel good about solving a problem that the expert on TV couldn’t solve.

Their reaction was unanimous. They identified with the guy and wanted them to give him his formula sheets. Some of them were pretty riled up about it!

They’re quite accustomed to me showing them videos and doing activities that are designed to build up their understanding that everyone approaches things differently, and we’ll all get there even if we take different paths. This guy wasn’t allowed to follow his path and do it his own way, and they were unfairly putting him on the spot and forcing him to do it their way.

It’s a rich problem, so I’ll use it again, but I think I’ll set it up and frame it a little differently next time!

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Dan Meyer

January 1, 2016 - 9:37 pm -

No doubt this particular tragedy has lots of authors. However, “he needed to see more worked examples” and “he shouldn’t have been allowed to use a calculator until the 12th grade” are particularly unsatisfying explanations for me. Neither addresses a crucial failing of this expert’s education:

This expert has come to the conclusion that math is a thing that requires formula sheets to understand and use.

Perhaps that’s just the result of the context of a deposition, which would raise the stakes and anxiety for anybody. But pursuing this theory, I’d recommend that teachers shouldn’t just ask students to calculate unknown dimensions using a scale factor. “Do I divide by the scale factor? Do I multiply?” wonders the anxious student, who believes the question can only be answered by a formula sheet or a math teacher.

Instead, give students one dimension and ask them to estimate the rest on the drawing, before they calculate. That may help them understand that their answer is reasonable, which will help them understand that math is reasonable, with or without a formula sheet.

2. #### Simon Gregg

January 1, 2016 - 10:50 pm -

The expert’s problem seems to be with the first stage of working out a decimal for that measurement 3 and 3/16 inches (though if he’s so poor at that, I wonder whether he’d be much good at anything else).

It makes me wonder, what kind of materials he would have back at the office that would help him out? Is there a table of conversion from fractional measurements to decimal? That doesn’t seem very likely. Or does he have a 3/16 key on his computer keyboard? I wouldn’t think so. Is there some kind of formula sheet for this? Or did he Google it or ask someone else in the office?

3. #### gasstationwithoutpumps

January 1, 2016 - 11:05 pm -
3 3/8″ at a 240:1 scale gives 67.5′ which rounds to 68′

It is easy to mix up 3/8″ and 3/16″, which is one reason I prefer doing measurements in metric units.

January 1, 2016 - 11:26 pm -

A few thoughts here:

The expert likely uses software to make calculations without understanding how they are calculated. This would be akin to me using an Excel mortgage calculator. I could find your monthly mortgage payments for you but I would need “my formula sheets” for that. I’m not making excuses for him, but certainly there is a line where it becomes socially unacceptable to not understand how a formula works. I think mortgage calculators are on one side and converting distance units is on the other.

This was certainly a high pressure and unsafe environment. It reminds me of times where I have to introduce people to one another and even though I’ve known everyone for years, under the pressure sometimes I will blank out on someone’s name and feel foolish. So, maybe there was an off chance he could have figured it out, but under those circumstances where anything short of perfection would be a huge issue, he decided to do nothing.

Lastly, it’s worth noting that eventually the heated conversation shifts from the actual math to whether or not he will do it or can do it. At that point it seems to become a pride issue.
5. #### Alex

January 2, 2016 - 2:48 am -

Based purely on the video (which is a reconstruction), the type of calculator might be significant.

Most people at school use scientific calculators, with a dedicated “fraction” button. When you press it they show a fraction with spaces for the top and bottom numbers. It makes it really easy to enter “3 and 3/16”.

The reconstructionist is given an office calculator, which doesn’t even have brackets. He needs to enter a counter-intuitive sequence of “3/16+3” to even get the starting point. When I was at school I remember being aware that most people wouldn’t be able to handle that kind of mental contortion. They’d never been asked to.

So what’s the problem, and how might we solve it? Well, the man’s been given the wrong tool for the job. He’s never been asked to use the wrong tool before & so this throws him. This makes him defensive and he latches onto an excuse about formula sheets.

But other than his pride, I don’t think too much has gone wrong here. In the future he should bring his own calculator (and laptop) to these meetings. He should also show his workings somewhere in his official report, so he can bring them out if needed.

6. #### Simon Gregg

January 2, 2016 - 3:59 am -

That sounds quite plausible, Alex. I’d not thought of calculators with fractions.

7. #### Simon Gregg

January 2, 2016 - 4:08 am -

But if he was an “expert”, I would have expected him to be able to get round that with an office calculator, even with a bit of nerves.

I also agree with gasstationwithoutpumps – my first thought when I saw this was: how much simpler this would be in metric!

8. #### Simon k

January 2, 2016 - 4:31 am -

I think the context is crucial here.
It reminds me of the times when a politician is asked an arithmetic problem e.g. 6 x 8. Quite often they refuse to answer. I honestly don’t think this is down to them not knowing but it isn’t worth the risk.

Also the calculator is very different tool to the spreadsheet.
If he wasn’t able to recreate it using the same tools he did in the first place, and without the legal teams breathing down his neck, that would concern me.
Though estimating is a great idea, it probably wouldn’t have caught the situation where he mis-typed and ended up with 75 ft. Would 7ft matter, possibly very much.

It reminds me a little of teachers who want their students to divide 987654.123 by 17 by hand.

9. #### Jo

January 2, 2016 - 5:49 am -
The advantage to only working with kids in the fourth grade and under is that I have to put every question through the “how could a fourth grader solve this?” filter.

So, 3 and 3/16 inches at a scale of 20′ = 1″.

3 inches at 20 feet per inch is 20 + 20 + 20 = 60. Each 1/16 of an inch is equal to 1.25″ (they might need a calculator for that as they work with remainders in division in fourth grade not decimals though the more advanced ones could learn it). 1.25 + 1.25 + 1.25 = 3.75. 60 + 3.75 = 63.75.

How do I know my answer is reasonable? 3/16 is a little less than a fourth. A fourth of 20 is 5. 60 + 5 = 65. 65 is 1.25 more than 63.75.

This is not a problem of having the wrong calculator.

10. #### Walter McGrain

January 2, 2016 - 6:31 am -

I get the sense there’s more at play here than what this purports to show. I know we’re supposed to walk away from this with the lesson that we shouldn’t rely on black-box formulas without knowing how to work out solutions from fist principles. But I think what is happening is that the expert witness does not want to set any precedent whatsoever that he’ll do any calculations on the witness stand which could result in errors while performing them under the pressure of a jury proceeding. He’s obviously being very careful in how he answers the questions from the lawyer which leads me to believe he understands the implications of that line of questioning.

11. #### Brendan

January 2, 2016 - 6:37 am -

Let’s make and assumption, as an expert witness he was really just a guy who does a job (he probably hates). He saw the opportunity of making a few extra bucks. He probably hasn’t thought about or cared about the numbers in his job for years. If he was lucky during his first week and maybe at a few PD sessions over the year someone talked about the actual math, but for the most part, day to day business means, he plugs and chugs numbers into spread sheet (Yes, fractions too because allowing the computer to convert those reduces the human error).
Back to the Math Wars. Do we teach understanding or teach doing?
Obviously he was more comfortable doing math. Given a situation he would choose the correct formula and solve the problem, but in this case he was unsure of the correct formula to use and thus refused to do the problem.
While he might have ended up with the right answer eventually, this was high stakes, where he was required to get the right answer the first time, and be able to explain each step along the way, while simultaneously solving each step. The option of estimating and or going back to fix mistakes was not there.
This was a PARCC test with the added stress of if you fail you become the laughingstock of your industry and you probably lose your job.
Could he go back and figure out the actual math behind his formula sheet? Probably, but that is a question he is not usually asked.
The defense lawyer did a great job of cross-examining the witness. Asking questions he knew the witness usually doesn’t need to answer in his real life job. Then he trapped him in a pride argument, where the expert had to admit he didn’t understand the real science behind the formulas, which of course he doesn’t have to in the normal course of his job. That would be like asking a carpenter to explain how to draw blueprints.

12. #### kate nerdypoo

January 2, 2016 - 6:38 am -

A majority of my students use calculators that allow you to input mixed numbers, so they rarely think about what 3 and 3/16 even means, sadly. That is a battle we’re all fighting, no doubt.

It’s quite possible he accidentally keyed in 6/16, which when multiplied by 20 gives 7.5, therefore giving 68 feet. This is also a reasonable error, since the 6 is directly above the 3 on the calculator.

This is why I always emphasize to my students that they should estimate the answer before using the calculator, so they can check if they’ve keyed something in wrong. I’d suggest to them to think of 3 and 3/16 as 3 and 4/16, which is therefore 3 and 1/4. 3*20 + 1/4*20 = 60 + 5 = 65. Therefore they know their answer has to be less than 65 and would recognize that their answer of approximately 68 must not be correct.

Thinking in teacher mode, I was struck by the fact that the defense attorney kept asking the question in the same way. He was really focused on turning 3/16 into a decimal, which in my mind is the least efficient way to handle most problems. Everyone wants decimals all the time but fractions are actually so much easier to work with, intuitively. I can’t turn 3/16 into a decimal in my head, and I also can’t multiply the resulting answer by 20 in my head, but I can really simply multiply 3/16 by 20! 3/16 * 20/1 = 3*20/16*1 = 3*5/4*1 = 15/4 = 4 and 1/4. Likewise, if I am tasked with taking the square root of 4/9, turning it into 0.4444444 totally obscures the answer of 2/3 (what?? the square root of 0.44444 is 0.66666??? I mean, that actually looks crazy!).

The defense attorney could’ve asked that question a lot of other ways that might’ve gotten the witness to recognize his mistake, but instead the tactic he took caused the witness to only dig his heels in and refuse to budge out of pride.

But actually the most disturbing part of this video is the (perhaps naive) realization that expert witnesses are paid for their testimony. Why is that legal? Does that not call into question their legitimacy, objectiveness, scruples, and impartiality?

13. #### Joshua

January 2, 2016 - 6:52 am -

The concluding notes in the video say:
>The motorcyclist’s lawyer filed a counter-motion to refuse >payment to the expert witness. It contained the math >standards for Wichita middle schools.

but also
>The counter-motion was not granted

Which seems to mean that the judge did not believe this exchange disqualified the witness as an expert.

What do you make of that?

14. #### Nathan H

January 2, 2016 - 7:15 am -

My guess is the witness measured the line on the print out without checking if it was scaled right. I print out maps for hiking and off-roading trips all the time. Almost never does the reported scale actually match what was printed.

Even so, the original measurements would have been made at full scale with the calculations based on those. And that would have made the printed figure’s dimensions immaterial. Lawyers make mountains out of mole hills and expert witnesses will play ignorant to avoid creating inconsistencies. Not at all uncommon for a deposition.

15. #### Jeff Nielso

January 2, 2016 - 8:20 am -
The motorcyclist’s lawyer is the unrelenting classroom didactic whose motivation is based on making his student look and feel stupid. I was waiting for Act 2 where the lawyer would jump up, grab his felt marker, and demonstrate just how easy he can show the procedure. Another victory for the teacher.
16. #### Evan Romer

January 2, 2016 - 8:26 am -
The NYTimes video Dan links to is an actor re-enacting the deposition. He uses the text of the deposition verbatim, but it is a re-enactment. A video of the original deposition is here:

(I haven’t watched the NYT re-enactment, so I haven’t compared the two.)

Evan Romer
Susquehanna Valley HS
Conklin Ny
(retired)

17. #### Dan Meyer

January 2, 2016 - 8:52 am -

Thanks for the comments, everyone. In particular error explanations from katenerdypoo and gasstations, learning explanations and other comments from Robert Kaplinksy, Alex, Jo, Jeff Nielso, Evan Romer. I’ve added them to the main post.

18. #### Bad At Math, Good at analysis

January 2, 2016 - 9:34 am -

You folks are so wrapped up in the math I think you’re missing the point.

This expert used his “formula sheets” to drive his testimony. Since he is basing his testimony on the answers provided by the “formula sheets” he *should* consistently refer back to them. If he does not, he could make himself look like an even bigger ass than he does here. If he failed anywhere, it was double and triple checking his work for mistakes.

Lawyers are very much into consistency, which is something I learned in writing software for a group of lawyers many years ago. Same input == same output every time as long as all data otherwise remains the same.

19. #### mmurph531

January 2, 2016 - 10:22 am -

I think the mistake starts with the need to make every fraction a decimal. Why does everyone use decimals when fractions seem to work so much easier in this case? to get an approximation First 3′ would equal 3×20′ or 60 feet using an approximation of 1/4 of 20 would give 5 extra feet…So he should have known that it is going to be less than 65′ so 68′ is way off.
and how hard is it to multiply 3/16 X 20? Sounds like 5th grade math…Really, no calculator is needed.

20. #### Jessie Turner

January 2, 2016 - 11:28 am -

No doubt this particular tragedy has lots of authors. However, “he needed to see more worked examples” and “he shouldn’t have been allowed to use a calculator until the 12th grade” are particularly unsatisfying explanations for me. Neither addresses a crucial failing of this expert’s education:

This expert has come to the conclusion that math is a thing that requires formula sheets to understand and use.

The comment regarding worked examples seems to imply that worked examples can’t convey conceptual knowledge and depth of understanding. The research on process-oriented worked examples shows that examples can be modified to improve conceptual understanding. Read http://bit.ly/1YXiSLP

21. #### Moschops

January 2, 2016 - 2:45 pm -

I’m not swearing in this post, because that’s a way to get a post wiped without consideration.

Blah blah blah “wrong tool for the job”, “made a mistake typing in the original work”, blah blah blah “why do we have to convert fractions to decimals” blah blah blah GET BENT

He was asked for the decimal version of three sixteenths, and said he couldn’t do it without reference material. That was the question. Asked very clearly. He was not asked “how did you do it originally”. He was not asked “can you repeat the steps you took originally, using whatever materials you used then, to repeat your assessment.”

Massively incompetent. No question. No excuses.

22. #### Andrew

January 2, 2016 - 4:30 pm -

For those saying, “he’s a peon,” “he doesn’t need to understand,” “he probably hates his job,” comparing it to flipping burgers, etc…

If you watch the actual footage, in the first minute he introduces himself as the president of an accident reconstruction company.

I’ll buy the not needing to know the ins and outs of the math for the average worker, but you’d think in such a litigious industry as accident reconstruction, someone at the firm would be able to explain the math behind the formula sheets. And if not the president, then hire someone.

I’m actually going to show this to my math sections and my business sections. Hiring the right people for the job is as important, if not more, as being able to do the job. Jack of all trades, hire a master.

23. #### Michael Caulfield

January 3, 2016 - 9:34 am -

I guess I see something else entirely here. In the actual deposition video you can see him punching in a few keys, I imagine he at least converted to a decimal. What if it started to dawn on him that something wasn’t quite right?

This is the moment we long for as educators, when the student realizes a current assumption or understanding is wrong and seeks to rectify it.

But you see what happens in an adversarial system that lacks safety for people. In this setting the realization that he might be wrong is horrifying and dangerous. He digs in, he stops trying to calculate because it might prove him more wrong.

I don’t think this is a bad person or a dumb person. I think this person is quite aware of how to convert fractions to decimals. But kick in the fight or flight reflex and all bets are off. The brain shuts down and figures out the most efficient way to short-circuit cognitive dissonance.

This is one of the reasons adversarial scenarios should be used in the classroom only when carefully designed. It’s probably also one of the reasons that the thesis — where you defend your point and demolish those of others is not the best foundational model of education. Peter Elbow used to talk (still talks?) about the need for Believing Games to counteract the effects of an unchecked adversarial system, I can think of no better advertisement for that than this. (Peter Elbow’s believing game: https://www.d.umn.edu/~cstroupe/ideas/believing.html )

This adversarial system might work OK in the courtroom, but when you see this behavior in business or the classroom you don’t have a faulty expert or a worksheet culture, you have a sick environment, too in love with a competition paradigm.
24. #### suehellman

January 3, 2016 - 10:00 am -

As I listened to the original deposition, what I heard was a man who made it clear that the formula he worked with not only calculated the distance but also handled any required conversions. As well, during his professional training he’d been given specific tools and set procedures for making this sort of assessment, and he trusted the judgement & choices made by the instructors at that level.

This is not unusual when adults train for new professions. Nursing students often get caught in a similar situation. An instructor makes a decision that a particular method of dosage calculation, for example, is THE way it should be done & accepts nothing else. This may arise because of what the research says about how best to ensure accuracy, or the choice may be rooted in how the instructor herself was taught to do these problems, or it may reflect the selection of a one-size-fits-all method that works across a spectrum of problems and can be picked up by students with a wide variety of math backgrounds in as short a time as possible. Instructors at this level are professionals in their fields and tend not to have the time, skill, knowledge, or will to champion flexibility, connect the dots from middle school math to the problems at hand, or fill in learning gaps that occurred decades before.

Adult learners can be even more reluctant to challenge such decisions than kids, and if their teachers are inflexible, then they learn to live & work inside the boxes chosen for them. Now with pumps pre-programmed to do all the calculations & conversions, nurses are even reluctant to question what they intuitively feel are incorrect results. So what do I think this man’s K-12 math teachers collectively might have taught him that could have helped him avoid this calamity? A predispoistion to expect errors and check results on a different day or in a different way. I think he probably did his steps & used his tools correctly but entered some bit of data incorrectly. If there’d been a policy in that office that no result went out unchecked by a different person because little errors can have big ramifications, the mistake would probably have been found, and the case would have remained focussed on who caused the accident rather than on the expert’s competency.

25. #### Steve

January 3, 2016 - 4:33 pm -

What if the problem isn’t a fractions one but a geometry one? I’ve watched it a couple times trying to figure out exactly what distance they’re referring to, and I’m not quite sure but I think it’s the distance along the arc that they want (the 68 feet). If you measured that with a ruler you could get 3+3/16″, which obviously won’t convert to 68′ because that’s not the correct thing to be measuring.

26. #### Sam Jones

January 4, 2016 - 7:45 pm -

The “reconstructionist” is earning \$900+ per hour for his testimony, as well as this deposition. In boxing terms I think it’s called a rope-a-dope.

27. #### mike

January 4, 2016 - 8:13 pm -

Dan you’re wrong in thinking this is a math analysis exercise, it’s very much a litigant exercise, and provides meaningful insight into the difference between math for which we have the luxury of being playful and inquisitive about, and math in which we must provide absolute answers for a specific reason.

It’s irrelevant whether the witness can or cannot calculate the question correctly, it’s whether it’s in the best interests of his client that he will provide any certain, meaningful answer at all.

And on that basis, he’s done a good job, considering he probably really doesn’t know how to answer the question.

Consider that HAD he provided a definite answer, it would invariably have been followed up with a series of increasingly complex calculations that would invariably at some point have rendered him unable to provide a mental, definite, and accurate answer. With this line of questioning in mind(something a witness must always keep in mind, not JUST the question asked at that moment), this line of questioning has no positive outcome for the client, and the witness is wise enough to know where the line of questioning is going and wise enough to know when to cut that line off at the head.

Mathematically I don’t rate him too highly, but as a lawyer, he’s definitely a smart witness and my man on the stand.

28. #### Christopher Danielson

January 6, 2016 - 12:48 pm -

A happy story…

I bought a bunch of hardwood yesterday evening. Unlike building lumber, hardwood is sold by the board foot—misleadingly a measure of volume: 12 inches by 12 inches by 1 inch.

Watching the man grind out the numbers for my chosen pieces of lumber was very very cool. Dude brought the full arsenal of tools to bear…tape measure, scrap wood as a notepad, office calculator, intermediate notes on paper, measurements in both inches and feet. Every bit of his work could be checked by the time he was done. Because my expectation and his final number were within about 12% of each other, I haven’t bothered.

29. #### Anna

January 12, 2016 - 6:05 pm -

Interesting note: my grade 7 math class is in the middle of our unit on fractions, decimals, and percents, so I showed them this video so we could work on the problem. I thought they’d get a chuckle out of it and feel good about solving a problem that the expert on TV couldn’t solve.

Their reaction was unanimous. They identified with the guy and wanted them to give him his formula sheets. Some of them were pretty riled up about it!

They’re quite accustomed to me showing them videos and doing activities that are designed to build up their understanding that everyone approaches things differently, and we’ll all get there even if we take different paths. This guy wasn’t allowed to follow his path and do it his own way, and they were unfairly putting him on the spot and forcing him to do it their way.

It’s a rich problem, so I’ll use it again, but I think I’ll set it up and frame it a little differently next time!