Picture this:

Detectives with a partially degraded/mixed DNA sample from a crime scene hit a roadblock in an investigation. The case goes cold. Months or maybe years later, someone breaks out that DNA sample and runs it through law enforcement databases, hoping to find a hit among previously convicted felons. Out of the million records, a hit! Case closed, right?

What I didn’t know before reading DNA’s Dirty Little Secret (mostly because it’s never come up on Law and Order) is that DNA samples are like fingerprints: analyzed based on how many “markers” the sample contains. (This fact about fingerprints, covered on Law and Order extensively, I knew.) The more markers the sample contains, the more complete the picture and the less likely the sample will match a person by random chance alone. Similarly, the fewer markers contained in the sample, the more likely that others will turn up as a match for that sample. And here’s where statistics comes in.

If run a suspect’s DNA against a crime scene sample and you determine that there’s a one in a million chance that this is not your guy, you may feel pretty confident you’ve solved the case. But, remember the scenario mentioned earlier: you had no leads and simply put a weak DNA sample into a database containing a million known offenders. The fact that you found a match isn’t all that much of a coincidence, in fact, it’s expected.

If your test is known to have a one in a million chance of matching someone else, and you run that test with a million random samples, you expect to get one hit due to random chance alone. In other words, we would expect one person in every million to match the sample, not just the owner of the DNA. One of those matches would be the owner of the DNA sample, but the rest are referred to as false positives: people who pass the conditions of the test despite not being the correct match.

As stated in the Washington Monthly article, the good news is that in cases where DNA samples have all 13 markers intact (are fresh, complete and untainted with other DNA samples), the odds of a false positive plummet to one in many trillions.

If a sample only contains 9 (out of 13) intact markers, the FBI considers the probability of a false positive to be approximately one in 750 million. Yet, an Arizona state employee apparently ran a series of tests in 2005 on Arizona’s state DNA database and found multiple people in that database of only 65,000 profiles who shared nine or more identical markers. Another test in 2006 found 903 pairs with nine or more matching DNA markers out of an Illinois database of 233,000 profiles. Of course, that’s not to say that 1806 people would show up as a match for any one single DNA sample. (We don’t know how many of those pairs share markers.) But it does mean that there are 903 random 9-marker DNA samples you could put into that database and get at least 2 matches.

This is why there is an altermative probability statistic called the Database Match Probability, which for a nine-marker match in a system the size of Arizona’s is roughly one in 11,000. Using that statistic, we would have expected approximately six hits for a DNA sample with nine markers. Obviously all 6 of them can’t be guilty, and there’s no guarantee that any one of them is guilty. How significant, then, is a “cold hit” in a criminal database? What if this were not only the way the police identified you as a suspect, but also their main (only?) evidence against you?

Unfortunately, according to the article linked above, the FBI rejects the use of this alternative statistic, either because it contradicts their own figures or because it makes evidence appear less compelling. Consequently, judges (typically ruling from precedent and not typically statisticians) have often deemed the Database Match Probability inadmissible in trials. In order to even get to a ruling, though, this of course assumes that defense lawyers are aware of, understand and are able to adequately explain to a judge this statistical concept in the first place, itself not a common occurrence.

I myself am only a casual statistical enthusiast, but I think I understand what’s going on. Here’s how I interpret things, based on some basic statistical reasoning. Feel free to point out any errors.

If you get a DNA match with someone from your list of suspects (generally a sample size of not more than a few dozen), you can be fairly confident that if the test itself is accurate, then you don’t have a false positive. The chances of someone whom you reasonably suspect in a crime (access, motive, M.O. etc.) matching a DNA sample and *not* being the perp is relatively small, and this is where the one in so many million/trillion statistics come into play. It’s not that you need a million or a trillion actual people to validate the assumption; rather, your sample size is so small that hits become overwhelmingly meaningful and the chances of false positives drop to near zero.

But, once you open your sample size up to millions (or even thousands) of random people, matches aren’t nearly as meaningful. Imagine being a jury on an Arizona case and hearing that there’s a one in 750 million chance that the guy on trial didn’t do it, when in fact we expect 6 out of 65,000 people would show up as equally likely suspects if you just ran the DNA sample through the database. That’s a difference of multiple orders of magnitude and really should affect how much weight you give to that DNA evidence.

And notice that the process is completely different, too: in one situation, we are testing one person to see whether he matches a DNA sample. Since the chances of any one random person matching any one random DNA sample are so small (even when the sample has fewer than 13 markers), a hit is a big statistical deal. In the other situation, we are testing thousands or millions of people, and anyone with basic statistical understanding expects the appropriate number of hits based on probability. I’m not going to say hits are meaningless, but simply expecting a match in this situation vs. not expecting a match in the former situation highlights just how different these processes are and how different the significant of a match really is.

In very simplistic terms, it’s the difference between

- knowing your killer is left-handed, and discovering that out of the four people who had access to the crime scene, only one of them was left-handed (still not one in millions kind of proof, but damning nonetheless) and

- just searching for left-handed people in the general population and using their left-handedness as evidence against them in this particular murder

And that doesn’t even take into account the fact that some people who identify primarily as either left- or right-handed in fact use their left hand for certain tasks (would murdering be one of them?) and their right hand for others.

I highly recommend reading the article linked above and discussing the statistical concepts as part of your math education, but must warn you that it does reference disturbing crimes. I would suggest that parents preview the article before using it instructionally. And please let me know if you enjoyed it as much as I did! (Well, except for the killing and all.)

A great site for families looking into “voting mathematics” is www.votepair.ca. The actual purpose of this site is to arrange strategic “vote swapping,” but the arguments presented in favour of this controversial practice (and the comments left by visitors to the site) really do illuminate the issues with our current election model.

While I’m neither encouraging nor discouraging the practice of strategic vote swapping, the mere fact that this practice exists speaks volumes about the practicalities of the current system, and of course, about statistics!

I thought I’d blog about this site because this is just one passionate, dedicated voter running the site out of his home who has made a list of how people could help him out.  One request was for:

Content. Need more engaging, visual content on voting reform and the pair process. Interested in posting stories that illustrate the problems with current electoral system as they surface in the media. Two have been posted on the blog, need lots more.

So, I figured that if your family ended up taking this voting math seriously and wanted to do some interesting research/analysis/models . . . then why not send it to Gerry for inclusion on his website?  It’s not necessarily about supporting his cause; it’s about an educational experience, and educating the public.

Gerry also asked for help with:

 Vote pairing process. Need to finalize exactly how this is going to work. I’ve got a plan, but can use some feedback. Need to get data on strategic voting ridings. Also concerns about phony registrants to think about.

What could be more mathematically delicious then designing a vote-swapping system?!  Tell him you’re a homeschooling family (or co-op) and would like to get the kids involved in this.  Gerry made a point of saying:

To get more involved, email pairvote@gmail.com and let me know what interests you. I’m more interested in a willingness to participate than in ability or experience. Heck, I’m learning lots on-the-job, and so can you.

Again, you need not agree with the goals of the site to get something out of the exercise.  It just occured to me that some families might be doing some really cool math work at home regarding the election.  If it can be put to good use and help voters make informed decisions, then why not?

Also feel free to leave in the comments below how you’re incorporating the mathematics of voting into your homeschooling studies.

This is a two-part series that puts forth the hypothesis that in our attempts to make math approachable and relevant to children, we have come to revere “concrete” math teaching tools when we should be focused instead on “authentic” learning experiences, many of which embrace the inherent abstract nature of mathematics.

Keep an eye out for it!

What could be more fun than making a pie chart, bar graph or map?  This site encourages readers to download templates for maps, graphs, charts, venn diagrams etc. and create your own visual representations of song lyrics, popular sayings or whatever’s on your mind

like this:

or this:

or this:

or this:

I hope you enjoy the site, and sharing the templates with your kids so they can make graphs, charts and diagrams of their favourite songs!

So, I’ll leave you with something that should keep you and your kids entertained for about a week or so.

The National Library of Virtual Manipulatives has a few very cool interactive algebra tools online.

My personal favourite is the equation solver, which is done by first setting up your equation by creating a “balanced scale” (I often wonder, do kids today even know what those kinds of scales are anymore, or have they become an anachronism?) and then performing mathematical operations to both sides to discover the missing value for x. Of all the interactive websites I’ve seen to visually represent the process of solving equations, this one stands out as my favourite. (If you have others you’d like to share, simply leave a link in the comment field below.)

You can choose the option to use negatives or simply work with positive numbers (two different links) so even if your child doesn’t yet have experience with negative numbers, you can let him/her loose on the site and see whether he/she is ready for this tool.

Also, be sure to check out “Stick or Switch” which is “The Monty Hall Problem” and an excellent introduction to simulations.

One caveat: I was thoroughly unimpressed with the “Function Machine” and I hesitate to recommend it simply because at no point does it ever reveal what the function in use actually was.  And, the two times I tried it, the function machine was using (only because I know how to figure out the function myself) first a quadratic function and then secondly an exponential function — not exactly easy for your child to discover on their own.  (The patterns, yes.  The actual functions, no.) So, use this as a number pattern recognition game perhaps, but unfortunately it will not help teach “functions.”  (How can one learn about functions if you never see the function in use by the machine?)

So, be sure to visit the Algebra equation solver, either with negatives or without negatives, the “Let’s Make a Deal” simulation, and whatever other puzzles from the list you find interesting.  I’ll see you in early February!

Recently, the Thomas B. Fordham foundation sponsored a study “Advanced Placement and International Baccalaureate: Do they deserve Gold Star Status?” to compare the academic merits of the two programs, specifically comparing the Biology, English, Mathematics and History offerings of each.

The very act of comparing the two, however, requires some judgement calls: since the IB program requires all students to take a senior math course, there can be as many as six different “levels” of math courses offered by Ontario IB schools. Obviously, the content and goals of these courses will vary widely. Certainly, taking the highest of the “Higher Level” (HL) offerings provides a very different foundation than the lowest of the “Standard Level” (SL) courses. Similarly, there are two AP Calculus exams: AB is meant to cover a half-year college course, while AB is meant to cover the material found in a full-year, first year, American college calculus course. The study chose to compare the SL curriculum to the AB exam, citing the justification that both programs are for students not continuing in a heavily math-based university program.

The first problem I have with the comparison is that students who are not intending to study math (or math-related fields) in university have the option of avoiding AP Calculus altogether. Yet, every IB student must take at least SL level math to graduate with the IB diploma. It’s perhaps a little unfair to compare a voluntary, advanced credit course with a required, high school mathematics program, but the point of the study was to establish whether one or both programs really deserved the reputation as a stellar academic program. So, we’ll go with it for now.

Given that comparison, I couldn’t comprehend how the IB math program scored higher (B-) than the AP calculus (C+). One of my beefs after several years of tutoring students in the IB program is how students are “pushed through” advanced math topics. This is because senior math is required of all students, and quite frankly, not all students are suited to senior level studies in math. And, given that your final IB “score” is dependent solely on the mark from the IB exit exams, an IB SL program really lends itself to teaching to the test.

When you know, for example, that there will be only one exam question concerning derivatives, probably based on throwing something into the air, it’s not too difficult to teach the least talented of students a pattern-based answer that requires no real understanding of the math concepts. (Remember, I taught you in about 45 seconds how to do it here.) There are few 12 year olds who aren’t capable of mimicking these steps.

Today’s update to the story comes via Jay Matthews at the Washington Post in Professor Says Editors Altered Review of AP, IB Courses:

David Klein of California State University at Northridge posted on his university’s Web site his original assessments of AP Calculus AB and IB Mathematics SL, which showed he would have given a C+ to the AP course and a C- to the IB course. The final version of the report, released Nov. 14, raised the IB grade to a B-, contradicting Klein’s view that the AP course was better
. . .
Klein says he does not consider either the AP or IB courses the gold standard for high school math, although in his original report he said they had some strengths not found in mainstream high school programs.
. . .
Klein also says that many of what he considered his strongest points were deleted by the editors, particularly his view that overuse of calculators could interfere with students’ mastery of analytical skills and conceptual understanding. (The report can be seen at http://www.edexcellence.net.)

This, I get. I’ve long taken issue with how both of these programs overuse calculators and minimize manual calculations. The standard argument in favour of using calculators to remove computational barriers is this: the kids can then focus on anaylzing and higher level thinking skills.

The problem is, as any good math student knows, our real understanding of mathematical concepts comes from using the underlying math, not avoiding it.

If you don’t know what dance the numbers are doing, you can’t possibly make meaningful sense of the result.

This is one reason why the MDM4U course (Data Management) is so hard for so many kids. I remember studying statistics at university before computer programs were used for the number crunching. I needed to know, by hand and with no formula cheat sheet, how to compute things like standard deviations and correlation coefficients.

You couldn’t get through that course without seeing all the nitty gritty steps involved in arriving at your stats. More importantly, since you knew exactly what you did with all the numbers, you understood what they meant. In the Data Management course, however, many students are using calculators and computers to avoid the “trivial” act of actually calculating the statistics — as if that somehow weren’t part of the point.

Granted, arithmetic isn’t mathematics, but arithmetic is how most of us come to understand numbers. Take that away from students, and they’re making a whole bunch of advanced conclusions, based on very little understanding.

I’m not arguing that it’s perhaps more interesting for lower-ability math students to be able to answer questions about whether given data shows a particular trend or correlation. But, without the ability to do all the work by hand, or at the very least understand it, these students are never going to be in positions where they’ll get to do that kind of higher level mathematical anaysis anyway. So, what exactly is this really preparing them for?

It would be one thing if these programs made it very clear that they are shielding the kids from a lot of the real work. I’m all for a full-disclosure statement that informs the students there is a lot more involved in doing this stuff for real, and that they should use their interest/success in these courses to decide whether they want to actually study these concepts (properly!) in university.

But the problem, and the point of this study, is that these programs are often heralded as models of excellence in education. That’s not exactly consistent with the “warning — we’re taking out the “hard stuff” so you can work at a higher conceptual level” disclaimer that should accompany these courses, at least as far as mathematics is concerned.

Public misunderstanding of mathematics and mathematical literacy (I guess the educrats want us to use the phrase numeracy now) have created a real problem in mathematics education. Because so many people think that math is “hard to do” they don’t realize how easy it actually is to teach and learn math through memorization of procedures and ignore understanding altogether. Therefore, to look at the “hard” questions on an AP Calculus or IB math exam and to see kids answer them looks impressive. Memorizing the encyclopaedia sounds impressive, too. And it is, but it’s a feat of memory and not of understanding or appreciation of knowledge.

For homeschoolers, the choice between the two programs is simple. It’s simply not possible to participate in IB offerings without regular enrollment at a local high schools, so AP is your only option.

Is AP worth taking at all? Yes, for many reasons. But, do so with the understanding that the act of preparing for the exam is separate from the act of learning calculus.

So many people didn’t know that below zero, larger numbers are lower and thus colder, that the lottery had to withdraw the game!

To quote one of the “victims”:

On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn’t.

I phoned Camelot and they fobbed me off with some story that -6 is higher – not lower – than -8 but I’m not having it.

Of course, much of the dispute was over the use of the word “lower” and one would hope that using the word “colder” wouldn’t have caused such a problem. This just shows that even when our minds know the application of a mathematical concept, for example that -10 degrees is colder than -8 degrees, we may still get the math wrong. This is why so many people think “word problems” are difficult: because sometimes the math we instinctively use in everyday life doesn’t register in our brains as being math.

Other applications of integers you can work into your math teaching at home:

• ALTITUDE: above/below sea level; going up vs. going down
• MONEY: balancing a chequebook; making a profit vs. losing money
• SPORTS: hockey players’ plus/minus ratings

Here’s a cute website with some online integer games and challenges. And, my booklet explaining operations with integers is freely available from my Teachers Pay Teachers website.

A few students have checked in with me about how their school years have started. It’s stories like these that make me grateful that I no longer have a vested interest in how teachers are teaching (or not teaching, as the case may be) my students:

Sarah, would you like to know how to mke the inverse of a parabola?? We take our paper, flip it over, turn it around and trace it! I turned to my friend and was like is this a sick joke? (Actual MSN transcript, edited only to make it a single paragraph.)

Now, some of you will have a mini-heart attack, although likely only if you’re a math tutor. The rest of you probably won’t get what’s going on here, so let me attempt to summarize in a non-mathy way. (You’re lucky, my first instinct was to explain the math. You can thank me later.)

Non-math summary: This student is in Gr. 12. This student is learning one of those things that most people only use when “building bridges.” This student is learning barebones tricks based on pattern recognition for putting something on the page that a teacher can mark as being correct without needing to understand any of the math behind it. And we’re wondering why bridges are collapsing all over the place?

To compare, I can teach you “calculus” in the same way right now, in about 12 seconds. Ready?

The derivative of 8x is 8. The derivative of 24x is 24. The derivative of -123x is -123. The derivative of 67x is 67. So, what’s the derivative of 17x? Not a trick question. It is in fact 17. Congratulations. I just taught you calculus. I can give you a test right now to prove that you know calculus and to prove to the Ministry of Education that I’ve taught calculus. You may think I’m kidding, but this is what passes for teaching in more classrooms than anyone wants to admit.

This is just another reason why we live here now.

I’ve always been interested in the mathematics of voting, perhaps even more so than in the election results themselves. A wonderful math activity is to learn about the upcoming Ontario referendum on electoral reform and then use past election data to see how or whether the proposed changes would have affected the outcomes of past elections. You can also find information on other voting systems (using a sample lesson plan with activity sheets or a web-based lecture for example) and discuss concepts such as strategic voting, fairness and representation.

I hope you have as much fun on Oct. 10, 2007 as I do!

As with any document like this, it can go out of date as policies change and it should not be considered your official resource. Always check directly with the university’s own website (often under “Admission” or “Prospective Students”) for the last word. Still, it’s a nice two-page pdf that is a handy reference guide for which schools will probably require which math course for 2008 admission, and was accurate as of November 2006. Note: the original info was collected from OUAC’s Electronic Info website.

Thanks, Wayne!