Sometimes, the Door Is Down the Hall

Today my blog is about one of the most important things I’ve learned as a teacher, and specifically as a teacher of math. I’m going to start with a story about a kid I tutored for a while, many years ago.

When I was younger and just starting to realize I had a passion for math and for teaching, I firmly believed that anyone could understand math and be good at it. Some people took to it more readily than others, but I was certain that given enough time and effort, every single person could excel.

Then I met Lief (not his real name).

Lief came to me when he was in grade 6, and our first tutoring session was about math questions involving time. The question we worked on was something like “Harry leaves home at 12:05 pm and arrives at his destination at 1:30 pm the same day. How long did the trip take?”. Lief was really struggling with the question, but I knew that I could explain it in such a way that he would not only be able to determine the correct answer he would fully understand how we did it and be able to answer many more similar questions. As my students would say, I have mad skillz when it comes to explaining math.

Boy was I wrong. I spent an hour with Lief and I used all my powers of teaching and explaining to no avail. Strewn about us were diagrams, pictures of clocks, number lines, a watch and even part of a model Volkswagen Beetle (don’t ask me why, I don’t remember). Lief just could not understand what the question was asking and why the answer was 1 hour 25 minutes (see how I threw that in there so you’d know if you got it right? 😉 ). I learned a valuable lesson that day.

Some people just aren’t wired for math. And that’s totally OK of course. Contrary to popular opinion, math is not a critical life skill. Aside from people like me I can’t think of a single person that needs to be able to complete the square of a quadratic function given in standard form in order to determine the coordinates of the vertex of the parabola. Proof? You most likely have no idea what in the world I was talking about there and I bet you do just fine. I know at least that you own some sort of electronic device capable of connecting you to the internet. That says something.

So why is this blog titled “Sometimes, the Door Is Down the Hall”? What am I talking about, you ask? Well you wouldn’t be the first to ask that. Allow me to explain.

Where I live in Ontario, Canada, students attend high school for four years — grades 9 through 12. During that time, in order to be awarded their high school diploma, they must successfully earn three credits in math. For most students, that means a grade 9, grade 10 and grade 11 credit, though some do grade 9, grade 10 and grade 12. It means that math is optional in grade 12, if all you want is a high school diploma. If you want a post-secondary education however, like college or university, you will most often need to take math in all four years of high school, and also be sure to choose the right math courses for your intended post-secondary program.

At each grade, there are choices of which math course to take. There are different paths and they don’t all lead to the same place. For example, a student who intends to study math or science at university will take what’s called “Academic” math in grades 9 and 10, then “University prep” math in grades 11 and 12. Conversely a student who intends to go to college to learn a trade will likely take “Applied” math in grades 9 and 10, then “College prep” math in grade 11 and maybe also in grade 12, depending on the requirements of the college program. Some students take what’s called “Essentials” math in grades 9 and 10, then “Workplace” math in either grade 11 or grade 12. These students are either not planning to attend a post-secondary institution or else are planning to go into a field that has nothing at all to do with math.

Phew! So that was kind of boring to read, right? But if you read it you may have noticed the glaring flaw in the system. A student must decide on a career path in grade 9. When they are 14 years old. Actually they have to pick their grade 9 courses when they are still in grade 8, so they and their parents have to make the call when the student is 13 years old. Who the heck knows what they want to do with their lives at the age of 13? When I was 13 I wanted to look at girls, play video games, eat steak as an afternoon snack and look at girls. And then look at girls. I had absolutely no idea what I wanted to do with my life. As a matter of fact now, at 43, I still don’t know what I want to do with my life (except for the looking at women part — I still do that and am lucky to have a wife that is exceptionally fun to look at). But I do know that I am happy with what I’m doing right now. Maybe that will change, maybe it won’t. But the key to my happiness is that I am doing something I am good at and that in turn makes me good at it. Read that last sentence a few times. It seems confusing but it isn’t. Try doing something you suck at for a long time. Keep telling yourself that you’ll get better if that makes it seem worthwhile. But I think you’ll find out that when you’re not good at something you are miserable doing it and then you are not good at it.

So how does this manifest in high school? Well it may seem obvious that since very few 13 year old kids have any clear idea about what they want to do when they finish high school — let alone as a career — that they choose the option that keeps all the doors open. In Ontario, that means that they will usually choose grade 9 academic math, because if they don’t they are closing the door to a post-secondary program that requires math. They do it again in grade 10, 11 and even in grade 12. I can not begin to count the number of students I have taught who have struggled mightily in math who then sign up for the hardest math course the following year because they don’t want to close doors. I’ll illustrate with an example. Good ol’ Hanz.

You might remember Hanz from “Steer With the Skid”. Hanz is a hard-working kid who does not have a lot of natural ability in math. By not a lot I mean he’s terrible at it. And please, before you object and say nobody who works hard can be terrible at something, look around. Some people are wired to be awesome at certain things and terrible at others. Some kids are born athletes, some are born artists, some are born mathematicians and some are born poets. You can improve your abilities in almost anything but that doesn’t mean you can excel in almost anything. Personally, as hard as I might have trained, I would never have been an Olympic sprinter. My legs are too short and I don’t have the reflexes. I’ve made peace with it.

So back to Hanz. Hanz doesn’t want to close doors, so he takes Calculus in grade 12. Currently in Ontario the course is actually called Calculus and Vectors. It’s the two hardest math topics in high school grouped together in one spectacular ride. Hanz has been miserable in math class ever since grade 9. He works hard, and puts in the time, but the most he can muster are grades in the 60’s, and it eats him up. His hard work is constantly rewarded with what he considers to be mediocre grades. He’s miserable because he’s convinced that he can’t be successful in life unless he’s successful in math and his definition of successful in math is marks in the 90’s, something he’s never been able to do. In trying so hard to keep a door open, Hanz has missed the fact that for him, there is no door marked “math”. He can’t see that if a program requires Calculus and he takes it and earns a 51% he won’t get in anyway. It’s a fruitless exercise. Yet every time I talk about this with Hanz or his dad Franz, they both insist that Hanz has to stay in Calculus so that he can keep his doors open. That’s when I shake my head and say “Sometimes, there is no door. Walk down the hall.”

See, if Hanz could recognize that there is no “math” door for him, he would be compelled to walk down the hall and see what other doors there are. If Hanz would spend more time in situations where he has natural strength, he’d know what those doors are and what lies beyond them, and he’d be so much happier. Unfortunately it’s extremely difficult to convince Hanz of this, and he spends all his energy working at something he was not wired for, spiraling further and further into self-loathing and often depression. I’ve seen it many times. I’m not exaggerating.

Now please, before you go off wondering how I can call myself a math teacher and be so willing to write kids off, understand that’s not what I am saying. I teach all levels of math. I am just as happy teaching someone like Hanz how to plan a family budget and the evils of credit card interest as I am teaching him how to take the derivative of a sinusoidal function that has been composed with the square of a logarithmic function in order to determine the instantaneous rate of change on the curve at the place where it intersects with a given exponential function. I work just as hard either way, and my reward is always Hanz’s success. It just pains me to see kids like Hanz convinced that they will end up “homeless under a bridge” (this is a saying my students have when they decide they are going nowhere in life) if they can’t do the derivative question. Honestly though, how many people can? And why on earth would most people need to? The derivative question is an exercise in abstract thought that is beautiful in its way, and critical for people going into a field where they have to solve high level math or science problems all the time, but it’s not the definition of intelligence or success.

Students like Hanz often ask me what courses they should choose when they are picking for the following year. I always say the same thing.

Me: “What are you good at?”

Student: “Well I’m good at <insert non-math or science discipline here> but that doesn’t get you anywhere so I need to take <insert completely inappropriate math or science course here>.”

Me: “Why would you take something that makes you so miserable?”

Student: “Because I need it to be successful. I need to keep my doors open.”

At that point I generally ask them how they intend to become successful in a field that requires them to be good at something that makes them miserable. They really never have an answer for that. Except for the door thing. My advice then is for them to take courses they enjoy, and that they excel in. Happy people who excel at what they do are always successful. Find one and ask them. You’ll see what I mean.

Thanks for reading,

Rich

Steer With the Skid

When you learn to drive in Canada, one of the most important lessons is what to do when your car enters into a skid. It’s not a question of “if” really. In Canada, it’s definitely “when”. Usually it will happen in your first winter of driving, so you’d better be prepared. The technique is to steer with the skid. It means that you have to fight your natural urge to steer in the direction you’d like for your car to go, and instead actually aim your wheels right at that concrete median your car has suddenly decided to crash into. Then, once you and your car agree on the direction you want to go, you gently steer the car away from disaster. It works, and in fact it’s the only way to handle the situation, except perhaps throwing open your car door and launching yourself out onto the pavement.

The physics behind why it works are fairly straightforward. When you enter into a skid your car has momentum which is carrying it in a direction that is usually not conducive to healthy living, and there’s nothing you can do about it because the friction between your wheels and the road has suddenly been reduced significantly by ice, water, gravel or some other non frictiony substance. This means that gross corrections where your wheels are pointed at an extreme angle to the skid won’t work, because the momentum of the car is overcoming the minimal friction at the wheels. So by pointing the wheels in the direction of the skid you force the momentum to cooperate with your goal of non-disaster, and then make relative small corrections which work because the little bit of friction you do still have is only slightly off from the massive momentum. Baby steps of correction eventually get you out of trouble. And it happens pretty quickly, as anyone who has ever done it can attest to. When you don’t understand the physics, it almost seems like magic.

The reason it has to be taught though, is because it’s so counter-intuitive. Aiming in the wrong direction so that you can go the right way feels like slowing down so that you can speed up. As it turns out, this driving lesson is actually an incredibly important life lesson as well. It shows up in so many ways. I’ll illustrate with a few examples of skids.

Skid 1: The Determined Daughter

Imagine this scenario. It’s 5 minutes before you have to leave the house to walk your daughter to school, and she is insisting that she does not want to wear a jacket. The problem though, is that you happen to know it’s 3° Celsius outside. If your daughter is anything like mine and she has her heart set on not wearing a jacket then you have 5 minutes to fight and win World War III. Good luck. This, ladies and gentlemen, is a bona fide skid. Your car is careening into two kinds of barriers. Frostbitten Child and Late Slip. Solution? Steer with the skid. Instead of fighting the momentum, agree that she does not need a jacket, but bring it with you and leave. If you really want to be fancy, don’t wear yours either. You carry hers and she can carry yours. Or you carry them both — it’s not really important. Once again if your daughter is anything like mine she will have a true aversion to discomfort, which is something she’ll start to feel within about 20 steps. She’ll ask for the jacket. If you didn’t wear yours, you can ask her for yours before she asks for hers. That will make it okay for her to ask for her own and still save face. Skid averted. No frostbite and no late slip.

Skid 2: The Quadratic Quandary

Here’s another scenario that I often encounter at school. Let’s say I’m helping a student solve a quadratic equation (apologies for this if you don’t know what that is — feel free to skip this part). Take this one for example:

x² − 5x − 14 = 0

If you’re a math teacher you know that a goodly portion of students unused to solving quadratics are going to try and isolate the variable the nice old fashioned way. You also know that it won’t work – totally destined to fail. You might be tempted to intervene before they try, and suggest a different method, but if you do then in the back of their mind they’ll always be wondering why not just isolate.

The best strategy pedagogically is to let the student try it. Agree that isolating the variable is a good plan. Help them with the operations – steer with the skid. As you work with them to isolate the variable generally this will happen:

x² − 5x = 14

At which point you can have a very valuable discussion about why we are stuck. The student may try some fancy footwork here, but thanks to you being on their side, you can navigate it with them and they’ll see that there’s nowhere to go. Then you can gently steer them toward other options. What do we know we can do with quadratic expressions? Factor them. So what? Let’s find out. I won’t get into the actual solution here, because it’s not important right now and in any case if you were following until now I’m fairly confident you can finish up. But for those who need to know, the solution is x = 7 or x = −2.

Skid 3: The Perturbed Parent

Once more this scenario is one I encounter as a teacher, but in fact it generalizes to any customer service industry. I actually really learned this well in my previous life as a software engineer when I would spend quite a bit of time on the phone with our users who would call when they were struggling with our software. Readers who are teachers will understand this situation pretty well. It goes like this:

Hanz is a student in your class who has written a math test for you and earned a fairly low grade – say, 54%. Hanz has plans to go to university (or college if you’re American – here in Canada college doesn’t mean quite what it does in the States) to become a doctor. Hanz needs a high school average of 91% to get into medical school. Thus the 54% on your test is a somewhat sub-optimal result. The next day you get a call from Hanz’s father, Franz. Franz opens the conversation by informing you that he is a lawyer, and that he has a real issue with the mark you gave Hanz on the test. Franz tells you that Hanz is extremely gifted in math and has always earned grades in the 90’s until your class. Hanz worked extremely hard preparing for the test and his tutor guaranteed that he was ready to ace it. Franz concludes that the whole mess is therefore your fault, because you are an unfair marker, a bad teacher, a horrible human being and quite possibly a chronic hater of children. Franz insists that you raise Hanz’s mark so that it is consistent with Hanz’s abilities and also consistent with his goal to become a neurosurgeon.

At this point it is incredibly tempting to get defensive, or be offensive. After all, Attorney Franz has attacked your professionalism (unfair marker, bad teacher) and your motivation for being a teacher (child-hater). Furthermore, if you know Hanz you know that “gifted” and “math” are not two words that you would put together in a sentence describing him, unless you could liberally sprinkle said sentence with the words “extremely” and “not”. However there is nothing to be gained by this response. All it will do is exacerbate the situation.

Instead, steer with the skid.

First, tell Franz that you understand why he’s upset. In fact you are upset by the grade as well – who wouldn’t be? Ask him about the hard work Hanz put in to prepare. Commiserate with Franz about the difficulties of watching young people work so hard and then not have it pay off. I am not being facetious here, and neither should you be in a situation like this. Put yourself in Franz’s shoes. Hanz worked hard, and hard work is supposed to equate with success. So why didn’t it? You and Franz can discuss this question. You can provide Franz with some questions to ask the tutor about the work he does with Hanz. You can recommend that Hanz come and see you to go over the test to see where the disconnect was. After all, since Hanz is so talented in math, there must have been a disconnect. When Franz sees that you have no intention of fighting him, his momentum joins yours and you can then steer him in the direction he needs to go, which is ultimately to realize that you did not “give” Hanz his mark – Hanz earned it. And getting to the bottom of why he earned a mark as low as he did is what you both want so that you can both help Hanz. This will ultimately help Franz see that it is Hanz who was at fault, and will also eliminate the need to address some of the more insulting parts of Franz’s opening tirade. It is entirely possible that during the conversation Franz will admit that the “marks in the 90’s” comment was not completely true, and referred to 2 quizzes Hanz wrote when he was in the 3rd grade. By the end of the conversation, Franz will know that you are on his and Hanz’s side, and that the energy of all three people is channeled in the same direction – not the direction of the skid anymore!

There are countless other scenarios I can come up with, all of which I have experienced personally (no, I never taught anyone named Hanz …), but the theme is always the same. A situation arises and the temptation is to fight against it, but fighting only escalates the problem. The solution is counter-instinctive and often requires strong self-control but pays huge dividends. Leverage the momentum of the skid for a quick and successful course correction.

Works in cars, works in life. Steer with the skid.

Thanks for reading,

Rich

Switch the Side, Switch the Sign?!!?

You can always judge my level of incredulity by my combination of punctuation. Two exclamation marks bookended by two question marks is a high level indeed. It’s the DEFCON 1 of incredulity. It comes from the way I’ve seen a lot of my students conditioned in algebra. It’s extremely sad. I’ll explain, but I have to start with a story about something that happened last week.

Thursday night I came home from play rehearsal and got the debrief about the household goings-on from my wife. Turns out my daughter, my 11-year old angel of happiness, was crying for something like 2 hours while I was gone. Dads out there with daughters (and I guess daughters with dads?) will know that there’s a special thing going on between dads and their girls. I think the best quote I ever read to describe it was this one:

Certain is it that there is no kind of affection so purely angelic as of a father to a daughter. In love to our wives there is desire; to our sons, ambition; but to our daughters there is something which there are no words to express.
– Joseph Addison

So when I find out that my daughter was so miserable, a part of me curls up in the fetal position and cries too. But you want to know what made it worse? What made it even more upsetting? The source of her pain was … algebra. Algebra! Her first algebra. A person’s first exposure to algebra should be special. It should be life-defining. It should be a cherished memory that warms you on command. A polished jewel of contentment forever residing at the center of your soul. And yet here’s my daughter, crying for 2 hours, and it was because of algebra. And I wasn’t home.

Talk about two ways to break my heart. Unacceptable!

So I dug a little deeper. My wife told me that my daughter was doing question after question, getting them right, and crying that she didn’t get it. My wife, no slouch in the math department for sure, was explaining the process, but for some reason it wasn’t getting through. My daughter just kept insisting she didn’t get it, all the while getting questions correct. How does this make sense? In what universe can a child keep getting questions right and through tears insist she doesn’t understand? The answer is because she was just following orders. The old “Switch the side, switch the sign” gambit. Consider this question she was working on:

x + 3 = 7

x = 7 – 3

x = 4

Correct, right? And easy too? So why was she so miserable?

It’s because she had no idea why she was doing what she was doing, or what any of it meant, and certainly no way to tell if her answer was right. She was right by accident, and because she was following a bunch of rules. In short, she wasn’t doing any math at all. She was, for all intents and purposes (or for all “intensive purposes” if you’re one of those people who hears sayings but never sees them written down) a trained monkey repeating a task. And nobody wants that … except for maybe the monkey because they get a lot of rewards for stuff like that … but no human wants that. And my daughter is exceedingly human.

You see, here’s what she was taught (or at the very least, to give her teacher the benefit of the doubt, it’s what she thought she was taught): To get the variable alone you have to move the number to the other side. If it’s plus you do minus and if it’s times you do divide. If it’s minus you do plus and if it’s divide you do times. Ouch. So much wrong with this I don’t even know how to start. But I do know that it’s taught this way so often that I have students who think that’s what algebra is. And I know of one teacher who used to have her students repeat the mantra “Switch the side, switch the sign.” I used to ask those students what they do if it’s multiplication or division. They said that’s when you don’t switch the sign. Ooooookay then.

So here’s the thing. When students are taught “rules” for solving equations they are not learning math. They are learning algorithms. An algorithm is a sequence of steps you follow to complete a specific task. You don’t need to know why, and in fact the reason algorithms are so powerful and so common is because you don’t need to know why. Long division is an excellent example. Many of us remember how to use long division to get the answer to 97654 divided by 7. But how many of us know why it works? The algorithm was designed to turn humans into calculators so that mathematicians didn’t have to do the tedious work. In WW1 there were literally rooms full of people – called calculators – who would do repetitive tedious calculations assigned to them by codebreakers. The codebreakers knew why the calculations were required, but the volume of work to do it was so vast that if the breakers themselves were to do the work they’d never decode a single message. So calculators were invented. They were people. A lot of them. And they were good at algorithms. But they didn’t know any math. Of course nowadays we have little computers that do the same job, but the concept is the same. A computer doesn’t think – it only follows instructions. It is excellent at executing algorithms. BUT THAT ISN’T MATH!!!

Ok. Back to the algebra, and my daughter. Saturday morning we were sitting in the family room, still in our pajamas, and Phineas and Ferb had just ended. The time was right. I told her I was going to help her with algebra but first she needed to forget everything she’d learned thus far. She happily agreed and put the misery in some invisible incinerator. Ahhh. Square one. Then we had this conversation:

Me: “Imagine I split your class up into groups of 2. Pick a partner.”
Her: “Alyssa!” (this was meant to be obvious to me)
Me: “Ok, now every pair has to pick one person for the blue team and one for the red.” (blue is her favourite colour – I’m not a rookie)
Her: “Blue!”
Me: “Ok, we’re going to play a game. Blue team goes first. Here’s the game. Pick a number but not a hard one. Don’t tell red team what it is. Now your job is to give Alyssa one hint, and if she gets it right you both get a point. Otherwise nobody scores.” (yeah, lame game, I know – but there are points and a blue team so she’s right on board)
Her: “Ok. My hint is it’s my favourite number.” (I saw this coming a mile away, and had a plan)
Me: “Right, Ok. But here’s the thing. You want to be SURE Alyssa will get it right. What if she can’t remember what your favourite number is? You want to give her a clue that will work for sure. And no using your number in the clue!”
Her: “It’s my birthday.” (Ha! I saw that coming too)
Me: “What if Alyssa forgot your birthday?”
Her: “How could she? She’s coming to my party!”
Me: “Good point. But what if she thinks your birthday is on a different day than the party? After all it is. You want to be sure she’ll guess your number, so make the hint foolproof.”
Her: “Ok, I get it. My hint is my number is 5 less than 10.”
Me: “Awesome! Ok, I’m Alyssa. Is it 5?”
Her: “You knew that already because you know my favourite number.”
Me: “Good point. Ok, here, it’s red team’s turn. If you increase my number by 7 you get 12.”
Her: “Hey you can’t pick the same number as me!” (Success!!!!)
Me: “Yes I can I can pick any number I want. Ok your turn.”
Her: “If you cut my number in half you get 10” (ooooh, nice one)
Me: “20?”
Her: “Yes! Ok, give me one now!”
Me: “If you multiply my number by 2 and then add 1, you get 13.”
Her: “6?”
Me: “6? Why 6?”
Her: “Because 6 x 2 is 12 and 12 + 1 is 13.”
Me: “Nice! Ok, you go.”
Her: “If you add 67354 to my number you get 90543.”
Me: “Ummm, are you going to know if I got it right?”
Her: “No. But you always get it right so I want to know the answer.” (Love her – I always get it right? She needs to talk to my wife!)
Me: “How am I supposed to get it? Those numbers are huge!”
Her: “Just do 90543 minus 67354!”
Me: “Right. I knew that. Ok it’s 23189.” (I rock at doing subtraction in my head – blows her away every time – she checked with a calculator)
Her: “Yes. Nice one Daddy.”

This went on for a long time. She really liked the game. At some point she realized that I was not keeping score and she got mad at me. Then she realized that it would always be a tie so she said it was not a good points system. We spent some time coming up with a better points system. She came up with something. It was fairly convoluted and had to do with blue team’s ability to do an aerial cartwheel so I lost, but I’m comfortable with that.

Then I told her a story about a dude from a long time ago named al-Khwarizmi, who wrote a book called “Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (no, I don’t have the name of the book memorized – I always have to look it up – but I always know the al-Jabr part). I said the name of the book is a pain to say so a lot of people just called it al-Jabr. I told her that al-Khwarizmi’s book was all about ways to answer questions like the one in the game if you were not good enough to do it in your head. Take for example the clue

“If you increase my number by 7 you get 12”

al-K (that’s what his peeps called him I bet) would have said the clue slightly differently. He would have said

“Suppose thing, increased by 7 ducats, results in 12 ducats.” (take a moment to explain that ducats are kind of like dollars)

Then he would have said that you can solve it by reversing the increase, and conclude

“Therefore thing is 12 ducats reduced by 7 ducats, which is to say 5 ducats”

Then we did a few that way. I’d write “Thing, multiplied by 6, results in 18 ducats” and she’d write “Therefore thing is 18 divided by 6, which is to say 3 ducats.”

Now some people might not believe that you can ask and expect an 11-year old to use language like this, but I’ve never understood why people would think that. Speak to them this way and they will listen, understand, and respond in kind. It’s what we’re wired to do. It’s how we learn to communicate.

I should mention that at one point she asked me if al-K called his book al-Jabr because it sounds like algebra. I told her that algebra sounds like al-Jabr because of that book! That al-K invented algebra. She thought that was super cool but also wanted to know how I could know such a thing. She was amazed to find out that I studied some history in my life. Daddy points scored.

Ok. So this gets tedious right? My daughter agreed. It’s too much writing. So then I told her about a dude named Rene Descartes who really liked al-K’s methods, but was too lazy to write it all out that way. So he’d look at the sentence

“Suppose thing, increased by 7 ducats, results in 12 ducats.”

And he’d say for example that “thing” is too many letters to write, but it’s important since it’s the number we’re trying to get people to guess. So Descartes chose the minimum number of letters possible. One. I let her choose the letter. She chose “m”. She always chooses “m” when letter-choosing is the task at hand. Then I told her that “increased by” is a pain to write out also, and asked her what she thought Descartes would say instead. She wrote down “+”. Then I said what’s “results in”? She wrote “=”. And Voila! She had written

m + 7 = 12

Then without me saying anything else, she said “Oh, so then Descartes would write m = 12 – 7! Which is 5!”.

And honestly, with that the lesson was done. I gave her about 8 more equations to solve in the Descartes style, and she got them all. We never once discussed rules, and we never once switched a bloody sign.

And there were no tears.

Maybe I’ve rescued the algebra memory.

Thanks for reading,

Rich