# Monthly Archives: February 2017

I remember the moment well – it was almost bedtime, and my son, already in pajamas, asked me about the derivation of the quadratic formula. Yeah, yeah, that’s kind of thing just happens in our family. Since it had been a while (like 25 years) since I had studied this myself I looked on the internet for the best way to present the idea of completing the square.

I loathe Khan Academy, but decided it might be useful for this kind of refresher. Boy was I wrong! The opening of the video started off ok as he discussed the fact that quadratic equations that are perfect squares are super easy to solve.  Check! If we can make any quadratic look like its a perfect square we can solve that easily too. Great! I won’t bore you with the details, but needless to say it went downhill from there as the video immediately dived into a bunch of manipulations and never once explained things conceptually. You see, completing the square is actually really easy to understand if you actually draw a square. Hey look, we’re taking a rectangle and cutting off some bits to make it a square and adding another little bit to finish it off! You can draw it out and show the student visually exactly what’s happening. Here’s an example of a site that gets it right.

This is the crux of my problem with Khan Academy – a lot of procedural mathematics with little focus on understanding. It might be ok for test prep review when concepts have already been mastered, but I absolutely cringe when I hear of homeschoolers using it as their primary math program or parents of gifted kids using it to accelerate. These kids need to be challenged, inspired, and taught problem solving skills, not memorization of mindless algorithms. For a child that is behind or already turned off from math, seeing numbers fly around the screen with no idea of the underlying concepts could lead to even more confusion. Real math is not about manipulation, its about knowing how to think and set up a problem.

I do think there are some good videos out there for learning math, for instance AOPS has some that follow along with their booksEducation Unboxed has outstanding teacher training videos for using Cuisenaire rods, and The Great Courses has some cool lectures that we’ve really enjoyed. Additionally, online courses can be a great way for motivated older students to learn. However, when it comes to elementary and middle school math in particular, I really believe that human interaction, discussion, hands-on manipulatives, and instant feedback are essential components of learning how to think about numbers. For a more detailed and articulate critique of Khan Academy I highly recommend this article from the Washington Post: Khan Academy: the revolution that isn’t. The title says it all.

Filed under Educational philosophy, Gifted math, Uncategorized

## Review: The National Museum of Mathematics

When it comes to museums I’m pretty picky. I wasn’t expecting much when I planned on visiting MoMath, the National Museum of Mathematics, during our trip to New York City two weekends ago. Their website is pretty minimalist and few people have even heard of this museum, but it turns out that MoMath on 26th street in Manhattan is quite a gem. My son was delighted at the idea of a whole museum devoted to mathematics, and the giant “Pi” handle on the double glass entrance doors just raised our expectations of what was inside. There were two floors of interactive exhibits and tons of friendly staff to help explain and interpret the math.

A few of our favorites:

The bicycle with square wheels

My son heard about this concept a while ago, so he was super excited to finally see one in person and try it out. There’s a great explanation of how this works here.

Enigma Cafe

I love science museums that have an area to sit down and explore small puzzles and games at your own pace and this one was especially well done – nestled in a cozy wood paneled area of the bottom floor. The puzzles themselves were highly engaging and tactile but with small screens to explain the goal, solution, and mathematics behind each.  If only it had been a real cafe and served coffee, I would have been in heaven.

Tessellation wall

Younger visitors to the museum gravitated to this station which had a huge wall with different types of magnetic tessellations. We loved the fact that the interactive panel nearby offered challenges appropriate for older students and we enjoyed the chance to build our own designs.

The cryptography machine

This was a natural fit for my son who has been studying cryptography since he was in second grade using CryptoClub.

Shape Ranger

This interactive light table used to explore the efficiency of shape packing was a favorite of mine.
Suggestions for future exhibits: I would love to see something focusing on cellular automata through the Game of Life or Abelian sandpiles (a concept my son and I are playing with in our homeschool). Also an exhibit to explain the Monty Hall problem would be a lot of fun. I feel both of these lend themselves easily to cool interactive explorations.

My only real critique of the museum, and of a lot of museums actually, is that they require you to stand and read a lot of background information on the nearby displays to really get the most out of the exhibits. For example on the bottom floor they had a wonderfully done plinko machine with marbles dropping in to demonstrate binomial probability distributions. This was great for my son, who has been exposed to this idea many times, to see as a live demonstration. However, for an elementary student who is just learning about this concept, a 15 minute read-through at a kiosk is asking a lot when surrounded by so much excitement going on around them. The Harmony of the Spheres exhibit proposing to connect music and math was similarly distracting – kids touching the balls to light up and make sounds without really learning much. From a cognitive science standpoint I would love to see more museums integrate the learning into the exhibits by requiring predictions and other user feedback rather than having explanatory text that is separate from the physical demonstration.

At the same time I do have to give credit to MoMath for putting a lot of thought into the information they present. Rather than overwhelm users with a lot at once they offer enough for each visitor to get started and then offer a choice of more information for those who are interested.  They also have plenty of staff who help to explain and interpret the exhibits.

MoMath is a great time, especially for a math loving child. It is centrally located across from Madison Square Park and reasonably priced at \$15 for adults and \$10 for students. The museum claims to target children in the 4th-8th grade range, which I thought was very accurate though there is plenty for adults as well. The gift shop is a special treat with lots of math related games, puzzles, and art work. My son wanted to return and we will definitely make it a priority on our next visit to New York City.

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In my previous post I talked about how variety is a key component of math acceleration. But what does that look like in practice? For my last two years of elementary school I was lucky enough to attend an innovative full-time gifted program that did an incredible job of covering the basics while simultaneously doing creative project-based learning. Math for that program was split into two parts: 1) regular math class which was a standard program but entirely self-paced and 2) a math laboratory class which covered unusual topics like probability, different base number systems, graph theory, and cool projects like visualizing a million. I loved this idea so much that I decided to use it with my own student. On a weekly basis we spend approximately 2/3 of our time working in our main math book (currently Art of Problem Solving’s Introduction to Algebra) while another 1/3 of our time is spent on fun and interesting topics outside of the mainstream. A typical day might look like this:

Morning Main lesson: AOPS Algebra – 10 minute challenge problem from a previous chapter, 10 minutes of easier review exercises, 20 minute lesson on new material such as factoring quadratic equations.

Mid-morning break: 20 minutes working on a Scratch programming project that draws a line based on user inputed point and slope.

Most evenings:  30-60  minutes of discussion about Venn diagrams, logic, probability, triangle numbers, geometric drawing, playing with Zometool, or whatever strikes our fancy.  These discussions are led by me but not forced and usually start out with “Have you ever heard of…?”

I have a math-loving student and provide a rich educational environment at home so access to fun books, games, apps, toys, and videos are also part of his math education. All of this often adds up to 2 hours or more of daily mathematics, and it adds up effortlessly. Even for a student like mine who absolutely loves math, plodding through pages of standard exercises and working until mastery in every area would be an uphill battle. For students who are working ahead of grade level, keeping their chronological age in mind and being realistic about attention span is essential.  Changing topics, interleaving review with new material, and combining difficult problems with easier questions keeps attention and interest high which allows us to work for a longer period of time and cover more material. It also has the added benefit of helping the student make new connections. Despite what we’ve been told, mathematics is not a single lane track leading from arithmetic to calculus.

The idea of a math laboratory is also perfect for parents who want to supplement their child’s school work with something at home. After a long day of school and homework trying to pile on more math learning during evenings and weekends can be a tough sell to a worn out kid. However, if the topics are fun and unusual, it can be a natural and joyful activity. Most of my math laboratory activities are lesson plans that I put together myself, but a few of my favorite off-the-shelf math enrichment resources are listed here:

I plan to review some of these books and resources in the future, but all are excellent. With Scratch programming I like to challenge my student to make math related programs that review a previous concept or explore something interesting like finding prime numbers or perfect numbers.

The main thing is don’t be afraid to mix things up, head off course, backtrack, and follow those rabbit trails of learning wherever they might lead.  You’ll be surprised at how much material you can cover when you don’t stick to a single minded plan or curriculum.