Math is Friend, Not Foe

Today I’d like to talk about everyone’s favorite, least-favorite subject in school: Math

Math can seem very scary and complicated. And this feeling is very valid to have – math does get more complicated the more you learn. When faced with a problem like:one can really long for the days of “If Sally has 5 apples and gives away 3 how many apples remain?”.

But the beautiful thing about it, is that all math can be simplified. It builds up into complicated equations and problems, but it can always be broken back down to simple things!

 For example, if you know that sin(π/2) = 1, and how integrals work, you can solve this problem pretty easily – BUT therein lies the rub. Math is all just building blocks, but the key is knowing and sometimes memorizing what blocks to use (which is never fun). Just like it’s hard to understand 5 – 3 if you’re not familiar with counting, it’s almost impossible to answer that question if you’ve never been taught an integral or what sine is. 

People place so much fear, stigma, and weight on doing math that it becomes so intimidating. So when faced with something strange and hard seeming, they panic and assume “I can never understand this”. BUT THEY CAN!

We often don’t think about it, but back in elementary when everything about numbers and addition was new, it seemed so confusing then too.

When I was in high school, I once tutored a fifth grader who needed help with long division, and I found myself having to relearn some key concepts that I’d forgotten and hadn’t thought about in years! I noticed the student was adding the large numbers incorrectly as they worked, so I said, “Oh you have to carry the one here.” To which they responded: “Why?”

And I had to pause, and then think about because, why DO we do that? I knew it was something to do with numbers no longer being the same category but how did I explain that to someone unfamiliar? Which led me down a bit of a rabbit hole of looking up elementary school textbooks and lesson plans. Because without understanding something like “place values” (ones, tens, hundreds) it’s really hard to understand why you need to “carry a one” in long multiplication and division!

Even something as retroactively simple as division was once completely baffling – but once you learn it, it becomes instinct. The same can be said for the most complex of mathematics: once you understand its rules, nothing can stop you! 

If you can remember the rules of a triangle, you can remember sine. If you can understand a grid, you can build up to calculus and beyond.

Now the hard part is learning all of this for the first time – especially when many people have been burnt out from math, or set at a disadvantage in the past by bad explanations and inattentive teachers who never helped them understand. That is the true tragedy of math, and admittedly makes the subject a lot harder for most. 

But I’m here to say, never doubt yourself! Even old dogs can learn new tricks, and you can always learn the blocks and break it down when facing “complex” math!

Think like a gamer: there are challenges, but with enough effort, grinding, and practice you can overcome it (and there’s nothing wrong with coming back later)!

Math is all just building blocks at the end of the day: simple individual concepts on their own, but you can assemble them into complex structures – amazing but confusing, equations that can describe the way our world and universe work – that can be very intimidating. However, I think it’s very comforting to remember from time to time that even the most intricate towers are still made of individual bricks; and with enough time and effort, you CAN break it back down! 

Your journey starts here! Let’s talk!

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  Timothy Durham was convicted of rape when a test showed his DNA genotype matched the DNA recovered from the crime scene. Eye witnesses testified that Durham was in a different state at the time of the incident, but he wasn’t exonerated from his 3000 year sentence until follow up DNA analysis showed that the preliminary forensic analysis used in the trial was misinterpreted (Thomas et al.). DNA testing serves an invaluable part in our criminal justice system, but it comes with complications and bias that can lead to wrongful convictions and misinformed juries.  

Background:

Forensic genetics aims to identify the origin of a biological sample. It can be traced back to Karl Landsteiner who discovered human ABO blood groups. With blood groups, suspects could be eliminated based on their blood type. For example, if a suspect had type A blood and there was type B blood left at the scene by the perpetrator, the investigators could rule out the suspect. Unfortunately, the reverse didn’t work. If the suspect and perpetrator both had the same blood type, it can’t be concluded that the suspect was guilty. Blood identifiers evolved into 8 distinct categories which improved the ability to eliminate suspects, but could only conclude that (if the system matched) it might be the perpetrator or this person might be related to the perpetrator (Jobling and Gill). The field evolved in the 1980s with the discovery of Minisatellites, lines of repetitive DNA that don’t encode proteins. Minisatellite patterns in DNA create what we know today as the DNA fingerprint, and it has a much higher accuracy rate than previous methods (Augustyn et al.). Around the same time, a breakthrough for victims of sexual assault was discovered. In the case of assault, bodily fluids of both the victim and the assaulter can get mixed. A process called differential lysis was created which could selectively increase the concentration of sperm in a fluid sample thereby allowing for accurate DNA testing (Jobling and Gill). Forensics evolved to use single-locus probe testing after several years because of its ease of interpretation. It looked at specific minisatellites that had high variability between subjects. This was the first DNA-based method to be used in a criminal investigation. Two murder homicides from several years apart were connected using DNA evidence. The evidence was then used to exonerate an innocent man, and the guilty party was found when he tried to evade mandatory DNA testing for all men in the small town. His guilt was then reinforced when the DNA profiles matched, and he gave a confession (Jobling and Gill). Finding DNA at a crime scene doesn’t guarantee that you’ll find a murderer. There are issues with non-pure samples and limited samples that make DNA work difficult, and accreditation of facilities and new techniques and take time (Jobling and Gill).  

Subjectivity of DNA Interpretation:

In a 2009 publication, the National Academy of Science discusses the possibility for subjective influence in lab based forensics. The interpretation bias happens in complex cases where, for example, there could be a mixture of DNA. A study published in 2011 looked specifically at this issue. A gang rape happened in Georgia, and the mixture in DNA from the case was used to prosecute one suspect after another suspect took a plea bargain to testify against him. With DNA mixtures, there are three possible conclusions: suspect excluded, suspect cannot be excluded, and inconclusive. If the suspect is excluded, it means that there’s little to no possibility that his DNA is part of the sample. If the suspect cannot be excluded, it means that there’s a possibility that his DNA is part of the sample. In the trial, the DNA examiners who knew the context of the case concluded that the suspect being prosecuted could not be excluded. The 2011 study gave the DNA evidence from this case and the DNA profiles of the suspects to 17 forensic examiners. These examiners had no context of the case, and they all worked independently. Of the 17, only 1 came to the same conclusion (suspect cannot be excluded) as the examiners who knew the context of the case. Twelve of the 17 concluded that the suspect who was prosecuted and convicted had no relation to the DNA found in the sample. Not only does this show that subjectivity occurs due to the context of the case, but the 17 examiners came up with different findings based on their past experiences meaning that not all DNA analysis is objective. Some organizations like the Scientific Working Group on DNA Analysis Methods say that quantitative statistical evidence should be presented with every categorical conclusion, but the US doesn’t require this (Dror and Hampikian).

False Positives:

There are two categories that cause wrongful identification in a DNA report: coincidental matches and false positives. Coincidental matches are DNA matches between two unrelated people. The probability of these matches is about 1 to 500,000 for white individuals and about 1 to 350,000 for black individuals. This small probability of coincidental matches makes people feel confident in DNA evidence. False positives, on the other hand, are matches between DNA and test DNA that result from issues in the forensics lab. There have been multiple cases of forensics labs accidentally switching victim and suspected rapist DNA which lead to convictions that were later overturned when the lab realized the error. There are also cases of misreading or misunderstanding a sample. In a case from 1993, a man was convicted of rape because someone interpreted a mixed sample of victim and perpetrator DNA as an individual sample. The conviction was again overturned. To prevent false convictions, retesting DNA is allowed, but it’s not an efficient solution. Many people are pushing for statistical information on the probability of false positives, so that the jurors can make accurate judgements based on DNA evidence (Thomas et al.).  

Conclusion:

DNA analysis is a useful tool for the criminal justice system, but subjectivity can be an issue in forensic testing. To combat wrongful convictions, there are organizations like The Innocence Project who use DNA testing to free the innocent, but reformed forensic testing standards should be put in place to prevent subjectivity and to ensure juries are correctly informed during trials. https://www.innocenceproject.org/    

Citation:

Augustyn , Adam, et al. “DNA Finerprinting.” Encyclopedia Britannica , 26 Apr. 2018, www.britannica.com/science/DNA-fingerprinting#ref153186. Dror, Itiel E., and Greg Hampikian. “Subjectivity and Bias in Forensic DNA Mixture Interpretation.” Science & Justice, vol. 51, no. 4, 2011, pp. 204–208., doi:10.1016/j.scijus.2011.08.004. Jobling, Mark A., and Peter Gill. “Encoded Evidence: DNA in Forensic Analysis.” Nature Reviews: Genetics, vol. 5, no. 10, Oct. 2004, pp. 739–751., doi:10.1038/nrg1455. Thompson, William C, et al. “How the Probability of a False Positive Affects the Value of DNA Evidence.” Journal of Forensic Sciences, vol. 48, no. 1, Jan. 2003, pp. 1–8., doi:10.1520/jfs2001171.