Rimwe Educational Resources 

The pressure is on.  It's the first post in Rimwe's newly created blog.  There are so many problems to pick from, I'm not sure where to start. 

However, upon further reflection, I think I'll start with the problem your brain has with math.  In this case, the problem your brain has with math ... about MONEY.  (Cue the theme song from The Celebrity Apprentice)  Specifically, I want to talk about the problems with understanding basic finance concepts that I saw demonstrated in my (adult) students in last night's class.  Basically, the problems were caused by them not paying (enough) attention to (important) details.  Turns out, this is related to the human brain and how we think. Also turns out, it's a problem we all have and not just in math.  I'll get to that part in a minute, but first I want to tell you about the three problems I saw in my class that I also think are running amok in the whole world:

1.  People do not read carefully.

2.  People do not pay enough attention to detail.

3.  People often have intuition that is mathematically incorrect.

Next, a little snapshot of how class went so you can "see" how my observations led to these conclusions.

I began the lesson as I often do, with a warm-up activity designed to both determine current level of knowledge (or misunderstanding) and also to garner interest & enthusiasm for the upcoming lesson.  The warm-up activity I used last night is called "Which Pays?" and it looks like this:

As you see, this was not an original creation of mine, it came as part of a lesson by Addison-Wesley and I love it.  I've used it for years.  It's great because it does two important things:  (1) makes you read and (2) makes you think.  In my opinion, any lesson that accomplishes those two things is a stark raving success.  Before reading any further, please do those two things:  read the problem above (carefully) and think about the question asked.  Then, decide for yourself, if you were in Rubin's shoes, which offer would you take?  Please jot down your answer along with a one-sentence justification. Ok, now you can continue. 

Back to my class last night.  Just like what I did with you in the previous paragraph, in class before we begin "doing the math", I often ask my students for an estimation of the answer and for an initial guess about the answer -- I did that last night and asked my students to tell me which offer they would take.  Keep in mind, last week's lesson was all about exponential growth so they had previous exposure to the mathematics involved, just not in this particular money-related context.  Every person in the room chose the first offer.  The comments and explanations for this choice included:

• "$300 is going to be more than mere pennies, even if they double."
• "I would not work a whole day for a penny!"
• "Even on the last day, he'd only make 2^15 pennies"

I invite you to go ahead and solve Which Pays? before proceeding further in this blog -- email me if you want to share your solutions or ask for help.

After the students, working in groups, actually solved Which Pays?, we discussed what went wrong as, obviously, every single person in class initially made the wrong choice about which offer would be best financially.  The problems?  Reading ... and Details ... and their Intuition.  After much lively discussion & debate (all by the students I might add, I served as observer, questioner, and, sometimes, mediator and translator), one student pointed out the "key" which caused everyone to change their initial answer (my additions in parentheses to help add clarity),

Yes! In carefully reading the problem, one must recognize the important detail that to find the total amount made for the second offer, one must find the sum of all the individual day's earnings. Problem solved.  Rubin (and you) should choose the penny-doubling offer because he'll make more money for the same number of days' work.

The solution is counter-intuiative... it goes against what we naturally think should be true.  Our intuition actually leads us astray mathematically.  It happens a lot. How? Let me explain.

On to the rest of the lesson.  In it, my students were given a scenario about two people saving for retirement and then asked a series of questions.

Here's the scenario:

Click here if you want to see the entire activity.  

Like before, I invite you to do what I asked of my students.  Answer the following before proceeding:

1. How much of Kim's own money did she invest?
2. How much of Chris's own money did he invest?
3. Who do you want to be at age 65, Kim or Chris?  Explain why in one sentence.

In my class, the students correctly figured out that Kim invested $24,000 of her own money and Chris invested $48,000 of his own money.  Own money = $200 x 12 x T where T is the number of years they invested. $200 is the amount they are investing every month. 12 is because there are 12 months in each year.  For Kim, T=10 yrs and for Chris, T=20 yrs.  It "makes sense" that, if Chris is investing for twice as long (20 years as opposed to 10), that he's investing double the amount of money that Kim is.  Unfortunately, that's where the correct answers ended.

I went around the room and asked, "Who do you want to be?"  Every single male in the room answered, "Chris" as did 25% of the females.  75% of the females wanted to be Kim, but sometimes for the wrong reason.  Reasons and explanations included:

• "I want to be Chris, at the end, I'll have more money AND a sports car!"
• "I want to be Chris, I saved twice as much because I saved twice as long as Kim.  I'll have more money."
• "I want to be Kim because after I'm older, I'll probably have kids and so won't have money to put away.  I like the 10 years then stop idea."

Now, my students last night did learn the importance of reading and paying attention to details from the Which Pays? activity because they picked up on the key details in the problem; however, they were still being lead astray by their intuitions ... and a complete lack of understanding of the difference between simple interest and compound interest ... caused by a complete lack of the ability to connect the previous week's lesson about exponential growth with the current topic of finance.  The lightbulb did eventually come on with my class and, like with the previous activity, it caused everyone to pick the correct choice:  you want to be Kim.  Exponential growth is larger for a smaller amount over a longer period of time when compared to a larger amount over a shorter period of time.  We don't naturally think this is true, but it IS true. When the math goes against our intuition, we get the answer wrong.  Why?   

Because these observations from my class are mirrored by humans in general, and that's because of how our brain works.  How our brain works?  Yes, let me explain.

I'm currently reading a great book titled, "Thinking:  Fast and Slow" by Daniel Kahneman on my Kindle.  So far, it's been fascinating.  I've already added several quotes to my clippings.  My occasional additions for clarification are inserted in the parentheses.  I'll start with his descriptions of the two "agents" who run our mind and control our thinking:

I really love this image.  I'm picturing two secret agent types, both spies but 1 is the instant stereotype: in a dark trench coat all obvious and "spy-y" while 2 is the real deal: less obvious but who really gets the job done. 

System 1, in the trench, is the brain's agent that is in charge of "fast thinking" -- the one who knows "that tone" when it's heard in a voice, that recognizes anger by its walk, and that predicts that guy is about to cut us off in heavy traffic. He's quick, almost effort-less, and he's always "on".

System 2, dressed like everybody else, is the brain's agent that is in control of "slow thinking" -- the one that you call on when you have a real problem, one that requires effort, deliberate action, and complete attention.  She's slower, extremely focused, and  must be actively engaged.  She only comes on when you ask.

So, you ask, "Why do humans have trouble with math?"  Well, the author explains, it works kind of like this:  Generally, Agent 1 is in charge of most of our thinking (it's more effortless, like an auto-pilot for our brain); turns out, that's not so good for us when we want to do math:

I will insert here that mathematical reasoning is performed primarily by Agent 2 for the reason he just mentioned above.  So, why does your brain have trouble with math?  Because, most of the time, we've got Agent 1 running things.  He's got a quick answer for everything; however, often his answer is based on faulty reasoning, incomplete data, or downright wrong information.  We need to be slower when we do math.  We need for Agent 2 to take over.  She's got to step in and say, "Wait a moment, let's really look at this, shall we?" and then walk us, step by logical step, through the hard work of effortful thinking and problem solving.  That's what is needed to be successful in math... and money .... and life.  

Daniel goes on to reveal an interesting trait we humans have when it comes to our own thinking:  we think we know more than we really do. Our metacognition (i.e. thinking about our thinking) is perceived better than it actually is.  Here's how he puts it:

In other words, we can be led astray by our Agent 1 intuitions that are telling us that something is true, just like my students were led astray by their fast thinking that more money invested at less time was better than less money invested at more time.  This can cause us to give the quick answer (from Agent 1) rather than engaging Agent 2.  In other words, we think we are thinking, but really we are not; it's just the knee-jerk response of Agent 1, rather than the considered, deliberate effort of Agent 2. Fascinating stuff, don't you think?

So, what's the answer to this problem?  We need more slow thinking.  We need to engage Agent 2 in math class and in life.

Stop... and think about it.

The Solver Blog

Author:  Dr. Diana S. Perdue