It’s Week 2 of our What’s That About? series and we can’t wait to break down the math (wait, there’s math in Chapter 2?) and the quote that introduces us to the wonderful Mr. Pikake. Let’s skip the fluff and dive right in.
What's in this article:
Super Quick Recap
Marco meets his tutor who is nice but also a little strange. He learns there’s more than one librarian, now his worldview’s changed. Mr. Pikake’s an Enderman making numbers teleport to new worlds… Liam’s is convinced the tutor’s after Marco’s yummy brains. Peter is mean and rude which clearly explains, why Marco is determined to control the numbers and earn some power back.
Wow. Can't wait to read that chapter! 😉
“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them”
This quote is meaningful both from the perspective of Fourier and his mathematical contributions and the idea of the use of analogies in mathematics. Let's take a moment to focus on why analogies are a powerful tool in mathematics instruction.
The Differentiation Struggle
Ask any math teacher one of the biggest struggles they have in their class. We bet one of them is differentiation. This term represents the use of different methods, approaches, or contexts used to introduce or reinforce a concept.
The struggle in math is that each student enters with a different level of understanding from the previous grade. If we want to teach proportions, a student must understand the many concepts they learned (or were supposed to learn) before.
Here’s an example: We are at an arcade. I don’t think the teller gave me the correct change. I bought 32 tokens. I look to the sign, and it doesn’t say anything about 32 tokens! It only says that I can buy 60 tokens for $30. Using just this information, can I figure out how much I should have been charged for my 32 tokens?
There is a lot happening here. For a student to grasp this concept they probably need an understanding of equations and expressions; knowledge of fractions would be super helpful, and knowing how to solve an equation would be awesome. It is very rare that in any given class every student has a good enough understanding of these things to solve this problem.
Teachers are faced with this difficult situation every day. They have certain goals to meet but a good portion of the class doesn’t have enough background knowledge to move forward with the material. They try to provide the students who need fraction practice activities to keep them busy and get them caught up enough to move on; the kids that are on target might start the material, and often the advanced learners are just bored.
MathBait and Analogies
How does Marco the Great address this? Simple. We don’t. (Cue shocked noises from the crowd.) We know that every child picking up the book comes in with a different level of understanding. We can’t possibly cater to everyone. Luckily, we have a magic trick up our sleeves: analogies.
No matter a student’s mathematical understanding, they have impeccable knowledge of the world around them. They know about toys and video games and friends and Halloween costumes. We use this to our advantage. By connecting mathematical concepts to relatable and tangible ideas, students don’t need exquisite prerequisite knowledge to pick up on the concepts discussed in Marco the Great.
This makes Marco the Great not only a fun read but an extremely relatable way to view mathematics and see just how much it impacts everything around us.
This quote is just perfect for Fourier. One of his biggest contributions to mathematics is a way to write any function in terms of sine or cosine. This is quite remarkable and extremely useful mathematically, but moreso, it is interesting in terms of analogies. Fourier showed us how we can take something that seems totally unrelatable and find the magical strings behind the veil that connects everything together. Marco the Great and the History of Numberville helps students through analogies to see past the magical mirror that disguises our world and to discover the hidden mathematics that is a part of everything in our own reality. When students are able to see math in terms they already understand, the subject is much easier to grasp and enjoyable too!
Welp. This is a story chapter meaning we get to hear some interesting things, but no math concepts are directly covered. So, what’s there to talk about? Those little equations on the bottom of the page are interesting…
(If you don’t have the book yet, order now to follow along with the next bit)
The math problems included in this chapter have a few purposes:
(1) They are a nod to the incredible price Mr. Pikake is offering…only $30?! Did you notice this pattern?
These problems are all about patterns. We don’t expect students to know about solving proportions or cross-multiplying, or fractions, or variables. These are all topics covered later in the story! The goal here is instead to find and recognize the patterns that are all around us. Not only are their patterns in these equations, but there are also patterns in how Mr. Pikake speaks. Did you notice those too?
(2) To introduce students to mathematical notation.
Don’t flip through Marco the Great when you first receive it. The equations, expressions, and mathematical notation are enough to make anyone a little scared. They are intimidating! Our goal in placing these equations in an earlier chapter, one that is all fun and no math, was to help students familiarize themselves with this foreign language they are about to learn. To plant the seed of what will come in a friendly way where there is no pressure.
(3) To work their problem-solving muscles.
Mathematics is the art of solving problems. Any problem can be expressed mathematically. A problem with friends, a problem with the garden, a car problem, you name it, there is math in there somewhere. We stress prerequisite knowledge so much and we have scaffolded instruction in math that says we learn this first, then that next, that we often loose the wonderful art of problem solving. So how will students do when they haven’t been introduced to any of the knowledge we traditionally associate with these problems?
Let’s take a look at that example again: We are at an arcade. I don’t think the teller gave me the correct change. I bought 32 tokens. I look to the sign, and it doesn’t say anything about 32 tokens! It only says that I can buy 60 tokens for $30. Using just this information can I figure out how much I should have been charged for my 32 tokens?
While a question like can look very scary, if we change our perspective just a bit, we might find some very interesting results.
If we can buy 60 tokens for $30 how much must each token be? Noticing that 60 is double 30 is the key here. Look for the patterns. This means we get 2 tokens for every dollar. So, 32 tokens must be… $16!
Are you reading Marco the Great with your kids? For this chapter, let them try it out. Encourage them to search for patterns. Encourage them to reason and problem solve. You might be surprised!
Although Marco the Great and the History of Numberville doesn’t explore the 6th grade topic of proportions, our digital content Vinculum Virtuoso takes a deep dive into these ideas. You can join our free online classes at Kryptografima.com.
Marco’s feelings change drastically before and after meeting Mr. Pikake. Have you ever been in a situation where you are nervous or scared about something that will happen but afterwards feel very differently?
Read Mr. Pikake’s lines out loud. Try to mimic how the text describes his speech. What do you notice?
Marco describes Peter’s car as looking like its owner. What does that mean? Can you think of other examples where an object might look like its owner?
If Mr. Pikake charges $20 for a 60-minute session, how much would it be if Marco wanted a 2-hour session?
Spiral review! What is 8+9÷3×2+(6-4)?
Marco says he uses math in his video game. His sword has strength 18 and his axe has strength 12. If both tools will only hold up for at most 4 uses, what is a better pick against a zombie with 60 health? Why?
Join us for Episode 3 where we get our first peak into Numberville!