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In my last post I discussed how its ok if elementary aged students are not yet working independently. Reading problems out loud, working side-by-side with your child, or providing other scaffolding to help them get through their work isn’t dooming them to use you as a crutch for the rest of their lives. It’s totally fine and developmentally normal for kids and young teens to still need some help getting work completed.

At the same time, moving towards independence in completing work and some degree of self-teaching is important. Incremental steps towards this goal can be taken gradually, without pressure, stress, or constant nagging. Here are some of my favorite techniques:

1. Cut out the busy work!  While there’s a certain logic in assigning routine, easy, fill-in-the-blank style problems to complete independently, the boring and contrived nature of that work often lead to less focus and more goofing off than something that requires real thought. Cut out the busy work altogether, practice routine skills through games, and find bite-sized problems that are just the right level to engage independently for a short period of time.
2. Do your own work. Just like all of us children can get lonely or feel isolated when completing school work on their own. Sitting right next to them at the kitchen table and completing your own work (even if its just catching up on emails to a friend), not only provides a role model for the student, but also company and connection.
3. Make it the final thing. As a gradual transition towards more independence do the first problems together and then leave the last few to work on independently. This can give students the emotional support and confidence they need for the problems while gradually improving stamina to complete more on their own.
4. Pomodoro technique. This is a really simple time management technique where work is broken down into focused 25 minute intervals with a 5 minute break. I’ve found that some students work much more effectively on their own when they know there is a limit to how long they will sit at the table.  For students just starting out or who have attention issues, starting with 10 or 15 minutes at first might be more appropriate. The time can gradually be adjusted upward. A visual timer helps some students stay motivated, but others will find a timer distracting or anxiety producing, so keeping it hidden and calling out 5 minute intervals might be more effective depending on the temperament of the child.
5. One problem at a time. Some students find a whole page of problems intimidating, especially when asked to work through them alone. An incredibly successful technique for me has been to write problems on note cards and have the student work through them one at a time. It’s a little more work for the teacher, but working through a stack of problems one-by-one gives a sense of accomplishment after each card is completed. There’s also less chance for the students to get intimidated by a long page of work.

In the end, independence with school work is nothing to stress about. Small, gradual steps, with patience and a sense of humor can make this a natural process rather than something difficult or stressful.

The picture above is of my son doing a Venn diagram challenge problem in the first chapter of Art of Problem Solving’s Counting and Probability. It’s a tough book, and a tough problem, especially for a 10 year-old. He knew exactly how to do it, but had made a few careless arithmetic errors so I asked him to try again and be a little more careful.

He sighed: “Can you just read it out loud to me? I want to do it together.”

By “do it together” he didn’t mean give hints, tell him the answer, or hold his hand through the work. He just wanted to work through his thought processes out loud to someone else. So I read the problem out loud. Then I  listened and repeated some of his intermediate steps back to him. A lot of our math work is like this – me right by his side providing feedback or just a sympathetic ear. And that’s ok! I see a lot of parents get wrapped around the idea that their children should be working mostly independently at this age and they get disappointed when they can’t just hand their child a worksheet and have it done efficiently. It’s wonderful if your kid can go off and finish a few worksheet pages by themselves but there’s also nothing wrong if they can’t.

I would also argue that there’s such a thing as too much independent work. Human interaction almost always enhances learning, making math (or any subject) much more pleasant and also helping retention. The ability of a parent or teacher to provide instant feedback should also not be underestimated. Being able to correct mistakes immediately, provide guidance, or simply see where a child is at, are all easier when you’re sitting right next to them.

Working independently is an important skill that all students will need someday, but its important to remember that someday doesn’t have to be today! In my next post I’ll discuss some of the techniques you can use for getting students to work a bit more independently in a natural and gradual way.

Recently two books about math education crossed my path. The first is The Math Myth and Other STEM Delusions by Andrew Hacker and the other is A Mathematician’s Lament by Paul Lockhart. Although quite different in their perspective, each author questions the current math education being offered by traditional schools.

The Math Myth challenges the standard high school math sequence and Hacker questions the value of any math beyond Algebra I. He makes some interesting points about math being elevated as a subject simply because it is easily measured through standardized testing and how its used as a proxy for overall intelligence. He argues that few professions have any need for advanced algebra, trigonometry, or calculus and that requiring these subjects for college entrance acts as a defacto gate-keeping mechanism that primarily hurts poor and minority students who have less access to tutoring and math-rich environments. The author essentially asks: Why should someone be denied the opportunity to study for a career in graphic design or a job as a vet tech simply because they haven’t mastered the binomial theorem and other obscure math principles that they will never use in their personal or professional life? I found myself agreeing with some of Hacker’s points but I object to the idea that K-12 education is solely about practicality and that mathematics does not help one learn how to think.

A Mathematician’s Lament is also severely critical of the current state of math education, but Lockhart takes a different and more startling view – math is more similar to fine arts, he argues, and should be treated as we do drawing, painting, and music. These artistic disciplines definitely require technical skills and knowledge to do well, but unlike mathematics there’s also an emphasis on performance, creativity, and joy. He makes an analogy that how most students learn math now would be equivalent to teaching painting with a  “paint by numbers” program that only requires students to match colors to predetermined blocks on a canvas. In Lockhart’s world, math education would center around discovery, originality, abstraction, and beauty. He abhors the idea of making math more practical and relevant: “In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide — a relief from daily life, an anodyne to the practical workaday world.” He makes a strong argument for the kind of interesting puzzles and pondering that I see excites my students.

I see merit in both of these arguments, and one way to reconcile both viewpoints, that I strongly argue for, is a greater emphasis on discrete mathematics in our school system. Pushing every student into a math sequence that leads to calculus is inappropriate and unnecessary. Topics like combinatorics, number theory, logic, and graph theory are very approachable to students of varying ability, teach mathematical reasoning skills, and are just plain fun. So many of these topics can be framed as puzzles with elegant solutions which elevates mathematics into something more than tedious calculations. For example, my 10 year-old son and I are working through a book on group theory (the study of symmetry) and we just created some really cool diagrams of the group formed when certain rules are applied to switching art on 3 walls. There are no numbers involved, but its much closer to what real mathematicians do than adding up columns of sums.

From a practical standpoint discrete math is foundational to the study of computer science, and its no less “rigorous” (whatever that means) than calculus. I would also love to see a statistics course added to the high school sequence, especially one that emphasizes interpreting statistics, data visualization, and graphic design. Students going into any science related field, including social sciences like political science, psychology, and sociology, would benefit enormously from this kind of background and in today’s data driven world, making sense of data and newspaper graphics is part of being an informed citizen.

I don’t hold out hope that mathematics teaching in schools is going to change any time soon, but whenever I hear criticism of Common Core and a call for “back to basics” in math education, I inwardly cringe. I understand the frustration, but unfortunately there is no turning back. The math taught to our parents – routine memorizing of algorithms and focus on arithmetic computation has no place in a 21st century education. I welcome discussions, like those started by Hacker and Lockhart, that challenge the status quo, even if its not something I wholly agree with.

A while ago I made some recommendations for fun living math books that would inspire and educate students who have an interest in mathematics. Along those lines, here is a list of some of the videos we’ve enjoyed. Many can be found online at various websites or on Amazon or Netflix, but I’m leaving out links since availability changes quickly.

The Story of One  (2005) – This hour long BBC documentary about the history of numbers is my favorite! Presented in a humorous and highly entertaining style, it is perfect for kids ages 6 and up, but also educational and interesting enough for adults. The Amazon DVD is overpriced but the video is available at various places online and some libraries.

Between the Folds (2008) and The Origami Revolution (2017) – These fascinating documentaries are about origami as an art form with mathematical insights. I recommend for ages 10 and up, or younger if highly interested in origami.

Flatland: The Film (2007) – A very well done animated interpretation of the classic by Edwin Abbott. My son was riveted by the drama and concepts of this world. Available for purchase on Amazon and also online. Ages 7 and up.

The Secret Rules of Modern Living: Algorithms (2015) – Available on Netflix, this hour long documentary covers algorithms of various types, including traveling salesman, sorting, and other optimization problems. My son learned a great geometric representation of Euclid’s algorithm for finding greatest common factors simply from watching this show (his demonstration pictured above). Great for ages 8 and up.

Fermat’s Last Theorem (1996) – This documentary about Andrew Wiles, who solved one of the world’s most famous math problems, is absolutely gripping. The emotions and tensions involved and the peek into the lives of how academics actually work on problems makes for a great film for the budding mathematician. Recommended for ages 12 and up. Available at various places online.

The Great Math Mystery (2015) – A one hour NOVA documentary that discusses some of the mysteries of mathematics like finding Fibonacci numbers in botany, the connection between Pi and probability, and the idea that the mathematical underpinnings of reality might be proof that we are living in a simulation. This show is a survey of topics and thin on explanations but high on wow factor. Great for inspiring students to appreciate and find mathematical connections. Recommended for ages 10 and up. Available online and on DVD.

The Story of Maths (2008) – This four part BBC documentary about the history of mathematical development is a little more in-depth and serious than some of my previous recommendations, so younger students might find it dry. Its great for ages 12 and up who are interested in math and history. Currently available on Netflix.

The Code (2011) – This three part BBC series has a slightly sensationalistic style and explores how numbers and shapes play a role in the natural world. My son found the third episode on prediction the most interesting. Recommended for ages 8 and up. Currently available on Netflix.

Finally, no discussion about math movies would be complete without mentioning the 1959 classic Donald Duck in Mathmagic Land! Great for all ages and available online.

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.

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:

Mathematics: A Human Endeavor
Alien Math
Crypto Club
Leonardo’s Mirror and Other Puzzles
Time Travel Math
Patty Paper Geometry
Scratch Programming

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.