Block by Block Jr. is a 3d construction puzzle based on the classic Soma Cube puzzle. Like the Soma Cube, Block by Block Jr is a great way to learn to visualize shapes in 3 dimensions — an important skill in science, math and engineering. For instance chemists visualize molecules, mathematicians visualize data, and architects visualize buildings and other structures. Solving Block by Block Jr. puzzles also develops problem solving and teamwork skills.
Every year my daughter’s elementary school stages a day of hands-on science exhibits called Discovery Day. For this year’s event I decided to build a giant 3d puzzle, inspired by giant Soma cubes other people have built out of cardboard boxes. However, when I tested the Soma cube on kids, I found it was much too hard. Adults too had trouble with it.
So I developed a simpler version of the Soma Cube, with fewer, smaller pieces, which I call Block by Block Jr. The original Soma cube has seven blocks that form a 3x3x3 cube. Each block is made of 4 cubes, like a Tetris block, except for one block that is made of 3 cubes. Altogether the blocks contain 27 cubes, and can be assembled to make a 3x3x3 cube.
Block by Block Jr has six blocks: three blocks made of 4 cubes each, and three more blocks made of 1, 2, and 3 cubes. Altogether the blocks contain 18 cubes, and can be assembled to make a 3x3x2 rectangular solid. Block by Block Jr puzzles are definitely easier than Block by Block puzzles, but the hardest ones are still quite challenging. Here are the six Block by Block Jr pieces.
For Discovery day I built three versions of the puzzle: a giant version featuring built out of 10″ cubical cardboard boxes stuck together with duct tape,
a half-sized version made of 5″ cardboard cubes,
and a small version made of 1″ wooden cubes
In all three versions of the puzzle I stuck colored dots onto the blocks to make it easier to identify which block is which.
Here are instructions for making your own set, including where to order cardboard boxes and wooden cubes.
And here are challenge sheets containing puzzles that require 2 to 6 blocks. Caution: there are answers for only the first 5 (of 10) challenge sheets.
You are welcome to make copies of this puzzle for personal use or for gifts, but please do not sell them…I’m working on turning this puzzle into a product.
I often use this activity to start a class. Hand out one copy to each student. Students must then work in pairs to figure out the eight words. Few groups will be able to figure out all the words. Nonetheless students usually like the challenge, and enjoy working together. I encourage students to explore other shapes they can make by superimposing the designs beyond the ones I originally had in mind. Design 3, in particular, makes many different shapes. Each design, when assembled properly, makes a perfect inversion. There are three fundamentally different types of symmetries: 180 degree rotational symmetry, reflective symmetry about a horizontal axis, and reflective symmetry about a vertical axis. One word, #7, has to be turned sideways to be read.
MIRROR DRAWING Copyright 2000 Scott Kim. All rights reserved.
This is a game for two people. Each of you needs a pencil or a pen. One of you will lead, and the other will follow.
Leader: sit on the left. Put your pencil on your dot. Slowly start tracing the dotted arrow with your pencil. Keep drawing on your side of the paper. Draw anything you want. Move slowly so the follower can follow you.
Follower: sit on the right. Put your pencil on your dot. Follow the leader. Trace your arrow with your pencil. Keep following, as if looking in a mirror. Be sure your pencils are always the same distance from the middle line.
This fun, challenging activity is appropriate for young children all the way up to adults. It is trickier than you might think to mirror someone else’s motions, especially when they make curves and diagonals. Encourage students to draw slowly, so one person doesn’t get too far ahead of the other, and have students trade who leads and who follows from time to time. I have designed this handout to fit on an ordinary 8.5″ by 11″ sheet of paper, but it works better if you use larger sheets of paper. Large sheets of newsprint, available in art classrooms, work especially well. The resulting drawings are often quite beautiful. I often have students walk around the room and look at what each other has drawn, or post the results on a wall. When I give this activity in the context of inversions many students try writing words, which are quite difficult to follow in mirror drawing.
Here are some related activities.
TWO-HANDED WRITING.This activity makes a good warmup for Mirror Writing. Each student needs two pencils or pens, one in each hand, and a large sheet of paper in front of them. Notebook paper will do, but larger paper is better. Stand up. Following the teacher, move your hands in small circles in the air, being careful not to hit your neighbor. Follow the teacher as he or she makes different patterns in the air as if conducting an orchestra: circles one way, circles the other way, zig zags, smile arcs, rainbow arcs, and figure eights. Feel the sensation of moving your hands in mirror symmetry. Now lean over and let the tips of your pencil drag along the paper as you continue to move your hands in opposite directions. Don’t worry about what your drawing looks like; just enjoy the motion. If you are right handed, place both pencils at the center of the page. If your are left handed, place your pencils at the left and right edges of the page. Now with both hands, at the same time, write your first name in opposite directions. Your normal writing hand will write your name forwards and your other hand will write it backwards. This may sound impossible to do, but it is easier than it sounds. You may find that cursive is easier than printing. You may also find that it is easier if you don’t look at what you are writing, but instead focus on the movement of your hands. Don’t be disappointed if you are not able to do this, for some people this exercise is too hard. But many people find this exercise much easier than they expected. You can check your work by holding your paper up to the light and looking through the back of the paper.
DRAWING AROUND A POINT.This activity makes a good followup to Mirror Drawing. The idea is the same, except the two students draw in rotational symmetry about a point instead of reflective symmetry about a line. This is a game for two people. Each of you needs a pencil or a pen. One of you will lead, and the other will follow. Draw a dot in the middle of the page. Pretend this is a tree growing out of the page. Leader, place the tip of your pencil on the paper somewhere near the dot. Follower, place the tip of your pencil on the opposite side of the dot, and at the same distance, as if you were trying to hide from the other person behind the tree. Leader, slowly start moving your pencil. Follower, follow the leader, making sure your pencil always stays on the opposite side of the dot, and at the same distance. Leader, draw anything you want, being careful to move slowly. You can trade who leads and who follows if you like.
THREE-PERSON HANDSHAKE.Have students stand in groups of 3, facing each other. Groups of 4 are also okay, but groups of 2 are too small. It may help to push desks to the edge of the classroom or move to a bigger space. Ask each group to invent a handshake for all people in the group to do together in which every person does exactly the same thing. For instance, if one person croos right arm over left, then all three people must cross the same way. Have each group perform their handshake for the rest of the class. It is easier for people to see if everyone sits down except the performing group. You will be pleasantly surprised by the variety of handshakes that students invent.
This image is available as a poster by Dale Seymour Publications. There are 26 first names here, one for each letter of the alphabet. Students enjoy looking for their own names or names of friends. Some names are written in capitals, some in small letters, and some are mixed. Each lettering style occurs exactly twice, once as a boy’s name and once as a girl’s name. Names that start with letters at opposite ends of the alphabet — ANNIE and ZANE, for instance — appear on opposite sides of the center of the design. Every name is exactly symmetrical. Most have rotational symmetry, meaning that they look the same right side up Some have reflective symmetry, meaning that they look the same in a mirror. Sometimes the line of symmetry is horizontal; sometimes it is horizontal. Only the name OTTO has both rotational and reflective symmetry. Some students will want to try making inversions out of their own names. Of course some names are easier to invert than others. If a first name doesn’t seem to work, suggest trying a nickname, a last name, or a friend’s name.
Here is a complete list of the names on the poster and their symmetries. “H Mirror” stands for “mirror reflection about a horizontal line” and “V Mirror” stands for “mirror reflection about a vertical line.” The hyphenation shows how letters are grouped. For instance, in the name ANNIE, the first letter A turns into the fifth letter E, but the second letter N does not turn into the fourth letter I: instead, the middle NNI makes one indivisble chunk.
When I started my PhD project in 1981, the IBM PC was brand new, and it would be years before the Mac and Microsoft Windows would appear. Nonetheless I was familiar with graphical user interfaces because of my internship at nearby Xerox PARC, the visionary research center that spawned, among other things, the laser printer, windows and icons, and the bitmapped display. I was well-acquainted with Alan Kay’s vision of a Dynabook (he envisioned the notebook computer in 1975), and understood the power of graphic user interfaces and graphic tools like paint programs to bring the power of computers to artists and visual thinkers. At the time I was part of the digital typography program within the Stanford computer science department, which built computer programs for digital typeface designers. I was also enthralled by the Visual Thinking course at Stanford, which taught engineers in the product design program how to think visually. It struck me as odd, and deeply wrong, that we were building tools for visual artists in a programming language that was utterly symbolic, and lacking in visual sophistication. I yearned for a programming language that had the same visual clarity that graphic user interfaces had. So I set about wondering what a visual programming language might look like. If computers had been invented by artists and visually oriented people, instead of by mathematicians and engineers, how might they write programs? It seemed to me an important question, but one that hardly bothered most computer scientists. I read about a few attempts to build visual programming languages, and decided there was something fundamental I needed to understand. My journey took me deep into the foundations of computer science, where I asked fundamental questions like “what is programming” and “what is a user interaction” — questions that often get passed over in computer science (any definition of “programmiing” that starts with “a sequence of symbols that…” is not deep enough to encompass visual programming languages). I never did build a visual programming language, but I did built a rudimentary visual editor that demonstrated the idea of parsing structure from pixels, and laid out what I believe is the key for a successful visual programming language: the user and the computer must share the same visual representation of information. I’m posting my dissertation and the accompanying video demo to re-open the conversation. What in my dissertation is interesting to you? What other work do you know of along these lines? Do you know of foundational research in interaction design and visual programming? And most importantly, what would be a good next step to push the work forward? Please email me your thoughts. I’d love to hear them.
A couple weeks ago I read the best-selling business book Good to Great, by Jim Collins. I snagged one of the copies floating around Age of Learning, where I work. The book explains six key traits common to companies that have changed from average companies to stock market superstars. The book was published in 2001, so it has had time to seep into business culture. Although the study has its critics (see reviews on Amazon), I find the book inspiring.
I like best the Hedgehog concept. The clever fox knows many ways to attack the hedgehog. The hedgehog knows only one thing: to curl up and become a ball of spikes. But that is all it needs to know. Dogged adherence to a simple strategy wins the day. And so it is in business. Much to the surprise of Collins and his team of researchers, the overperforming companies he chronicles were not clever proud foxes, but instead unassuming hedgehogs who stuck to simple, almost simpleminded strategies, without fanfare or big re-orgs.
Unassuming yes, but also incredibly courageous. In the 1950s, the grocery chain Kroger, then half the size of grocery giant A&P, realized that consumers wanted superstores instead of small local markets. So it quietly went about revamping every one of its stores to fit the new model. Expensive, time-consuming, dogged, and ultimately winning. Meanwhile A&P, in love with its glorious past, hid from the brutal facts and instead flitted among many different strategies, eventually going bankrupt. Both were old established companies, but only one had the courage to face facts, and commit to a strategy.
Of course there are good strategies and bad strategies. Hedgehogs find their calling in the overlap of three circles: What your are passionate about, what you can be the best in the world at, and what drives your economic engine. In other words, you gotta love it, rock at it, and make money at it. And if you are not the best in the world at something, having the courage to drop it, as Wells Fargo did when it decided to give up global banking and focus only the western United States.
The most surprising thing about hedgehogs is that they slip under the radar. The CEOs profiled in Good to Great are unassuming, almost invisible, not interested in personal glory. Instead they focus on doing the right thing. They hire passionate people, listen to the market, and make a long string of consistent decisions with unwavering determination. Hedgehogs are not spectacular. But they produce spectacular results. Slow but sure wins the race.
Math education is a leaky boat that is engulfed in flames and sailing in the wrong direction. There is so much wrong that it is hard to know where to start. And fixing just one problem won’t save the ship — we have to address all the problems simultaneously.
Now that I’ve got your attention, let’s unpack the nautical analogy and see what we need to do to save the ship of math education. My goal is to lay out the full range of problems in math education, so we can decide where to act.
Broadly speaking, there are three problems with math education, which I equate with fire, leaks, and sailing the wrong direction. Here are the problems, and ways to fix them.
1. Poor pacing (fire). The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you’ll miss it lessons, with little opportunity to catch up. This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.
Here are three ways to fight the fire of poor pacing.
> Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are rather traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need — a reversal now known as the “flipped classroom.”
> Visual learning. I love the Kahn Academy. My high school aged son hates it, because he, like most students, is a visual learner, and Sal Kahn’s lecture stick largely to traditional symbolic math notation. Mind Research is a nonprofit co-founded by a dyslexic learner, Matthew Peterson, who sought to teach math without words. Mind Research now produces a full K-12 math curriculum that communicates range of math concepts through wordless games (yes algebra can be done without traditional notation). Not only do the games reach visual learners, the very lack of words causes students to want to talk about their experiences with each other, thus deepening their understanding. Math education needs to address all learners, not just kids who learn in words.
> Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, just now entering schools across the nation, boldly replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.
If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.
2. No meaning (leaks). Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful problems. Teaching math this way is like teaching the grammar and spelling of English without bothering to teach the meanings of words, or letting kids read books. No wonder the most common complaint in math class is “when are we ever going to use this?”
Here are three ways to plug the leaks of meaningless math.
> Use math. In our increasingly digital society, kids spend less and less time playing with actual physical stuff. All the more reason to get students out of their desks and into the world, where they can encounter math in its natural habitat, preferably integrated with other subject areas. My friend Warren Robinett told me “a middle-school teacher I knew would, after teaching the Pythagorean Theorem, take the kids out to the gym, and measure the length and width of the basketball court with a tape measure. Then they would go back to the classroom and predict the length of the diagonal. Then they would go back to the gym, and measure the actual diagonal length. She said some of the kids would look at her, open-mouthed, like she was a sorceress.”
> Read about math. Before we learn to speak, we listen to people speak. Before we learn to write, we read books. Before we play sports, we see athletes play sports. The same should apply to math. Before we do math ourselves, we should watch and read about other people doing math, so we can put math in a personal emotional context, and know what the experience of doing math is like. But wouldn’t reading about people doing math be deadly boring? Not if you are a good story teller. After all mathematics has a mythic power that weaves itself into ancient tales like Theseus and the Minotaur. My favorite recent math movie is a retelling of the classic math fable Flatland, which appeals as much to my 7 year old daughter as to my adult friends. Here’s a list of good children’s books that involve math.
> Create your own questions. In math class (and much of school) we answer questions that someone else made up. In real life questions aren’t handed to us. We often need to spend much time identifying the right question. The simplest way to have students ask their own questions is to have them make up their own test questions for each other. Students invariably invent much harder questions than the teacher would dare pose, and are far more motivated to answer questions invented by classmates than questions written by anonymous textbook committees. Mathfair.com goes further to propose that kids build and present their own physical puzzles in a science-fair-like setting. Kids can apply whatever level of creativity they want. Some focus on art. Some on story. Others add new variations to the puzzles or invent their own.
If we plug the leaks of meaningless math, we will grow a generation of resourceful mathematicians who understand how to solve problems. But are we teaching the right mathematics?
3. Mathematics itself (sailing in the wrong direction). The mathematics we teach in school is embarrassingly out of date. The geometry we teach is still closely based on Euclid’s Elements, which is over 2000 years old. We continue to teach calculus even though in practice calculus problems are solved by computer programs. Don’t get me wrong: geometry and calculus are wonderful subjects, and it is important to understand the principles of both. But we need to re-evaluate what is important to teach in light of today’s priorities and technologies.
Here are three ways to update what we teach as mathematics.
> Re-evaluate topics. The Common Core State Standards take small but important steps toward rebalancing what topics are taught in math. Gone are arcane topics like factoring polynomials. Instead, real world mathematics like data collection and statistics are given more attention. As Arthur Benjamin argues in a brief TED talk, statistics is more important than calculus as a practical skill.
> Teach process. The widely used Writer’s Workshop program teaches the full process of writing to students as young as kindergarten. The process accurately mirrors what real writers do, including searching for a topic, and revising a story based on critique. We need a similar program for the process of doing mathematics. The full process of doing math starts with asking questions. Math teacher Dan Meyer argues passionately in his TED talk that we do students a terrible disservice when we hand them problems with ready-made templates for solution procedures, instead of letting them wrestle with the questions themselves. Here is my diagram for the four steps of doing math. Conrad Wolfram created a similar diagram for his Computer-Based Math initiative.
> Use computers. In an era where everyone has access 24/7 to digital devices, it is insane to teach math as if those devices didn’t exist. In his TED talk, Conrad Wolfram points out that traditional math teachers spends most of their time teaching calculating by hand — the one thing that computers do really well. By letting students use mathematical power tools like Mathematica and Wolfram Alpha, teachers can spend more time teaching kids how to ask good questions, build mathematical models, verify their answers, and debug their analysis — the real work of doing mathematics. And students can work on interesting real-world problems, like analyzing trends in census data, that are impractical to tackle by hand.
So there you have it, my assessment of the problems in math education. I’ve left out many important practical problems, like teacher training and funding. My point is that there are many problems to fix in math education, and that solving just one of these problems will not get us where we want to go. Let’s be aware of all the problems, and move forward on all fronts together.