Archive for the ‘Math’ Category

Should You Push Math and Science?

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Media pundits and policymakers have been telling us for years that we need to graduate more STEM (Science, Technology, Engineering and Mathematics) students, because our companies just can’t get enough qualified workers. I wrote a little bit about this perceived “Sputnik Moment” last year.

But now there is a new report by the Economic Policy Institute that blasts a hole in that story. After crunching all the numbers, it seems that the U.S. has more than enough STEM graduates. In fact, for every two STEM graduates, only one is able to get a job in his or her field. This seems to match the reports I have been reading over the years in the “American Society of Engineering Educators” newsletters, which suggest that many of our engineering graduates are having trouble finding jobs.

Percent of high school graduates going to college, graduating, and then entering a STEM job.
Source: Economic Policy Institute

This report also challenges the critics who say that U.S. schools are failing because our students don’t score as well as China, Canada, and other rivals on international tests. While it’s true that on average, U.S. students only score in the middle of the pack, some analysts say that this is a very simplistic and misleading summary. It makes a good sound bite, but raging over our seemingly weak performance completely misses other more positive information buried in year-to-year trends, socioeconomic indicators, and test methodology. In fact, the U.S. has a lot of highly qualified students who score in the top tiers of these types of tests.

The emphasis of the report is really on clearing up misconceptions about our STEM labor market as it influences foreign guest-worker and immigration policies, but I’ll let someone else fret over that.

What concerns me is the whole idea of pushing certain career fields on kids, even for reasons that seem noble on the surface. Whose interest does it serve? If kids choose a STEM field because everyone is telling them how desperately our country needs them, which implies job opportunities, and then it turns out not to be true, then those kids just lost out on 4 years of their lives which might have been better spent studying something they really care about.

Policymakers worry  that too many kids drop out of STEM fields while in college, but the EPI report claims that, in reality, more kids transfer in to STEM fields from non-STEM fields while in college, so the net result is usually more STEM majors at the end of 4 years than at the beginning. It seems that we are bemoaning a problem that does not exist.

Why do we give so much attention to engineering a work force that suits the needs of industry? If we only focused on what’s best for each student, I believe there will still be plenty of highly qualified and motivated individuals in every field, because all students quite naturally have different interests. With a student-led curriculum, the only thing we might have a shortage of is mindless submission.

It’s important to expose kids to lots of different things, including math and science, but they need to have the space and freedom to follow their fascinations, even if you can’t imagine how they would ever make a living doing that. If they later decide to pursue a career for monetary or security reasons, that’s up to them. Just make sure their expectations match reality – and not someone else’s agenda.

Teaching Kids Math – Overcome Math Anxiety with Games

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Textbooks and workbooks really are not the best ways to start teaching kids math. Even if your child is mathematically inclined, there is no need to pull out the textbooks until they are older and have had a lot of friendly experience with numbers and patterns already. It’s the same way with learning to read. Kids who have been read to often, see other people read, and are regularly exposed to the printed word have a much easier time learning to read than those kids who have not.

Board games, card games, and even video games help build up a child’s familiarity and ease with mathematical thinking before they ever crack a workbook. Don’t think a game has to be labeled “Educational” to be worthwhile. Those are OK every now and then, but the best games are the ones your child will want to play again and again. I’ll show you some of our family favorites in the following video:

The books and games I mentioned in this video are:

Books: How Math Works (out of print but still available used), Family Math and Middle School Family Math, and 101 Best Family Card Games

Games: SequenceBlokusCribbageSet, Mastermind, Rummikub

Full Disclosure – I am an Amazon Affiliate, but I mainly include the links to help you read more about the items I recommend.

Math for Social Butterflies

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Faces are very important to my daughter Emma, because faces have feelings and stories to tell. Math, however, is not important to Emma, because to her, it is coldly impersonal and meaningless. Word problems are OK, but abstract equations – forget about it.

Knowing her “right-brain” nature, I tried to ease Emma into math gently, with lots of manipulatives, friendly math picture books, and hands-on Montessori type activities. She was fine with all of this and seemed to understand the concepts of quantity, comparisons, and even basic arithmetic (if I have 5 apples and give you 2 apples, how many do I have left?). When she was about six, we started the Miquon math books and she found the Cuisenaire rods mildly interesting, but she balked at any visual representation of equations. The plus, minus, and equal signs all looked the same to her and she could make no sense of it. This made her furious. She threw up a mental wall and refused to look at any of it.

Ordinarily, I would have just waited another six months to a year to approach this material, but I had made the mistake of joining a public charter school for homeschoolers and they were going to be checking up on her progress. So I really wanted to get her over the hurdle of recognizing simple symbols. I found a very interesting book by Mark Wahl called, Math for Humans, which helped me invent a technique to help Emma. Since her natural strengths were interpersonal and artistic, I decided that math symbols would be less threatening if I gave them faces and personalities.

I created little 5-inch tall paper dolls with bases that allowed them to stand up on a table. The plus and minus symbols became princesses with the appropriate symbol boldly displayed on their crowns.  The equal sign became the queen, also with the equal sign on her crown. The two princesses each had a miniature basket (found at the craft store). The basic premise was that the plus princess liked to collect things and add them to her basket. The minus princess liked to give things away, subtracting them from her basket.  The equal queen’s job was to count the objects in their basket at the end of a transaction.

Of course, I had to embellish each story somewhat. So when faced with an equation like “2 + 7 = ” the “plus princess” would take her basket, filled with two lentils, and go collecting seven more lentils from imaginary bushes. When she came home, the queen would check her basket and discover that she now had nine lentils. To reverse it, the “minus princess” would take nine lentils in her basket, then distribute seven of them to assorted small toys attending the story. Throughout the process, Emma would keep up a running dialogue between all the characters and I would do my best to contribute voices and personalities.

With the paper dolls, it didn’t take long for Emma to overcome her fear of the symbols and she was able to remember what each one represented, regardless of the way an equation was written (2 + 7 = 9 or 2 = 9 – 7). After the first week or so, we put away the paper dolls, but she still needed to draw dots or other objects to help her visualize the problems. She came up with some very imaginative pictures and faces to dress up the numbers in her workbooks. Her margins were works of art! It made math time last twice as long, but I didn’t mind because it was the only thing that made it bearable for her.

The problem was when she attended third grade at our public school. Her teacher would not allow doodling on any of the math homework, so Emma was at a disadvantage. She could not answer any problems without some sort of visual representation. At home, she could use her Cuisenaire rods or scratch paper, but at school or during test time, she was miserable. We finally decided to bring her back home during the spring semester.

Learning to multiply and divide was another odyssey. We took it slow, with lots of hands-on games and activities. I made another paper doll with a times symbol on her crown. Instead of lentils, she collected unit rods of equal length. So if the question was “2 X 3 =”, the multiplication princess would either collect two 3-unit rods or three 2-unit rods. We always did it both ways to show that the answer was the same. The division princess would reverse the work that the multiplication princess had done, taking the rods in her basket and dividing them amongst grateful small toys. Again, we only used the dolls for a week or so, until Emma could remember what the “X” and “÷” symbol meant. After that, we depended on the rods for all of her math work.

Memorizing the times tables was also a long process. I found some flash cards that presented each of the multiplication facts as a story and that helped a lot. We also quizzed each other on our weekly hikes. I would ask Emma, “What is 6 X 7?” and if she answered correctly, then she would ask me a question like, “What is one million times two million?” She was always amazed when I gave an answer – those numbers seemed so impossibly big to her. I was amazed that she didn’t mind this kind of verbal quizzing. She liked the social aspect of it.

When Emma was about twelve, math really started to come together for her. She still didn’t like it, but it became easier for her to think abstractly. We were able to move along at a much faster pace, although repetition was important. On her own, Emma realized that she had to constantly review old concepts while learning new things or else she tended to forget that she had ever learned it. Instant feedback also helped. In seventh grade, she used the interactive 7th grade math DVDs available from the Teaching Textbooks company. It was so helpful to know immediately if she was solving the problems correctly. Then she didn’t have a chance to learn the wrong method. Unfortunately, Teaching Textbooks doesn’t yet offer similar interactive DVDs for any of their other courses. However, a phenomenal resource called does have interactive math videos where students can practice new concepts, and teachers (or parents) can track the student’s progress. Khan Academy is always adding new videos. Best of all is it’s free!

I don’t know how well Emma would have absorbed these math videos when she was little, although if I had access to them I certainly would have tried it. But if you have a child who struggles with the abstract symbols of arithmetic, try putting a face on them. It could be anything – princesses, dinosaurs, puppies, spiders or maybe superheroes. I can envision a similar game with toy trucks, gathering or delivering unit blocks depending on what symbol is pasted on the cab. Use your imagination! It might be just the thing for overcoming a child’s resistance to unfamiliar concepts.

Unfortunately, I threw away our original paper dolls many years ago. But Emma (now 15) was gracious enough to draw some new ones for you. If you have a child that might need a little help recognizing basic arithmetic symbols, just click here to download Emma’s paper dolls:

Math Paper Dolls

You may print them out on heavy cardstock and cut out (loosely – no need to get exact). Your daughter may even want to color them first. Be sure to accentuate the “+”, “-“, and “=” symbols!

Math Picture Books

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Math is everywhere in our lives, not just textbooks. Reading math picture books or storybooks to your kids will help to show them the friendly approachable side of numbers and patterns. There are TONS of these kinds of books, but I’ll just show you a few of the ones we used in this video. You can find a lot of these in your library or Scholastic warehouse sales.


The books I reviewed in this video are:

The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger

Sir Cumference and the First Round Table and Sir Cumference and the Dragon of Pi, both by Cindy Neuschwander (there are other books in this series as well)

Opt: An Illusionary Tale by Arline and Joseph Baum

Polar Bear Math: Learning about Fractions from Klondike and Snow by Ann Whitehead Nagda and Cindy Bickel

If You Made a Million by David M. Schwartz

Big Numbers and Pictures that Show Just How Big They Are by Edward Packard

Big Book of Time: A Magical Adventure through the Seconds, Seasons, and Light-years of the Universe by William Edmonds. This book is no longer in print, but I found a similar book here.

Incredible Comparisons by Russell Ash. I can’t seem to find this book online, so maybe it is out-of-print, but here is a similar book by Russell Ash.

Learning Math using Multiple Intelligences

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Howard Gardner first explained his theory of Multiple Intelligences in his book Frames of Mind (1983). Since then, educators everywhere have learned how important it is to discover how each individual student prefers to learn and solve problems. The eight types that Gardner has identified are: musical, logical-mathematical, bodily-kinesthetic, interpersonal, spatial, intrapersonal, linguistic, and naturalist. The key thing to remember is that math doesn’t have to be taught or learned the same way. Mark Wahl’s book, Math for Humans: Teaching Math Through 8 Intelligences, shows us how.

To order any of Mark Wahl’s books, just visit his website:

Note: I’m not an affiliate or connected to Mark Wahl in any way –  just a happy customer.

Math Practice for Kids who Hate Math

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If you want your kids to practice basic arithmetic, but they hate using textbooks or workbooks, try using resources from Scholastic Professional Books. These fun books are written for teachers and classrooms, but most can easily be adapted for homeschool use.

Learning Math the Wild, Authentic, and Rebellious Way

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Learning Math

Ouroboros by Michael Maier, image from Wikimedia Commons

My son found this article called “Lockhart’s Lament” online (available here) and raved about it, saying “Mom, you have to read this. It’s just like what you are always saying.” It’s true – I loved it, and I want to help spread Lockhart’s message far and wide. The great thing is that Paul Lockhart is actually a research mathematician and K-12 teacher, so he has way more credibility than this one little homeschool mom.

His message is this: The standard textbook-driven curriculum that passes for mathematics instruction in our schools is not mathematics at all. What our kids are being taught is only the dried up imitation of math. He uses the analogies of a paint-by-numbers vs. actual painting, and learning to read musical notation without ever playing music.

He says, “In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul crushing ideas that constitute contemporary mathematics education.” (p.2)

Why do we do this to kids? Lockhart believes that most math teachers and curriculum providers have the best of intentions, but they don’t even realize what true mathematics can be. It should be about solving real problems, not the phony exercises in a textbook with the demonstrated solution right next to it, but real thought provoking problems/puzzles with no given answers. True mathematics is more of an art form, he says. It should be fascinating, not repetitive regurgitation. The people who best understand true mathematics are mathematicians, not teachers. Ideally we would have mathematicians who happen to be very good teachers too, because not every expert is good at teaching.

Lockhart isn’t saying that kids shouldn’t learn the basic computational skills needed for everyday life, but that those sorts of skills can be taught without all the worksheets. His big criticism is that we mistakenly fixate on getting kids through the dullest possible presentation of Algebra, Geometry and other higher mathematics while simultaneously squelching any hope of inspiring students to delve deeper. He scoffs at the common argument given that students should learn Algebra and Geometry because it develops critical thinking. He would rather see kids doing their own thinking than regurgitating theorems and postulates for a test.

Tests. I think a big reason for our emphasis on equations and formulas is the effort to standardize curriculum and measure results. The type of math education Lockhart describes in his article would be very difficult to administer in large classrooms, and equally difficult to assess. And we all know how much school administrators like to assess.

I faced this problem with my youngest child. When we moved from Connecticut to Hawaii, I knew that she would have to take the mandatory state test for 4th graders and I didn’t think she would know enough of the material unless I started her on a textbook. It really bothered me, because up until then we had just been playing games, using lots of manipulatives, exploring patterns, reading books like “One Hundred Hungry Ants,” and doing art related projects like a life-sized chalk drawing of a blue whale in our cul-de-sac. She didn’t decide that she hated math until faced with a workbook. It didn’t matter which brand I tried, she hated all of them (although there’s a few new ones I wish I could have tried 8 years ago). I think it would have made all the difference if I could have stuck with my convictions and waited a couple of years to introduce some of these things, but the mandatory test forced my hand. Even then, there were things on the test we had never covered, like stem-and-leaf-plots, which made her feel like she wasn’t good at math.

I was SO frustrated! And worried too. Even though my instincts told me that it was OK to wait until she was older, I didn’t have enough experience or confidence to know it would be OK. My oldest son waited until he was twelve before picking up a textbook again, and at the time I still didn’t know how that strategy would work out. But now I know. He learned everything he needed to pull down a good score on the SAT, which is all he really cared about because he wanted to get into college. But this all is exactly the sort of thing Lockhart is lamenting. We should not be treating mathematics like a ladder that must be climbed from basic addition to calculus. We should be approaching it the way the Ancient Greeks did, as something fascinating to be discovered.

He writes: “Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.” (p. 5)

My middle son (the one who found this article) happens to love math. He dutifully worked through all his textbooks, but he always preferred the non-textbook activities we did. And he very often came up with his own ways of solving problems before he learned the traditional method. He even taught himself how to solve probability problems before he realized that anyone else had done it too. I never instigated any of this. It was just him, thinking and wondering. And now that he has taken chemistry and physics classes in college, the only thing that bothers him is having to show his work. He solves problems his own way (and he is usually always right), and hates to be slowed down by showing a bunch of steps he doesn’t even use.

My only quibble with “Lockhart’s Lament” is that he really leaves us hanging. He makes a very convincing case for overhauling our math curriculum in favor of something more wild and authentic, but since I suspect very few of us homeschooling parents are mathematicians, how are we supposed to recognize wild and authentic mathematical thinking? And just what are we supposed to do about it? I’m going to dedicate all my posts for the month of March to this question. I’ll show you the cool stuff I have found and used over the years (which I think Lockhart would approve of) and scour the math teaching universe for the most innovative ways to buck the current system.

10 Inspiring Websites for Learning Science

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Science and math don’t just exist in textbooks – in fact, the best part of these fields DON’T exist at all in textbooks. The curiosity, wonder, and magic must come first. Only then are we motivated to find out the details of how it all works. But sometimes it’s hard to show this to kids, especially if we never discovered an interest in these things ourselves.

As I wrote in my last post, it’s great if you can find a passionate, knowledgeable teacher or other mentor to lead a class, workshop, field trip, or other experience for your homeschool group. But if you can’t find teachers like that in your local area, the next best thing is to find the books they have written, or the websites they have put together. When I evaluate websites, I am really interested in the knowledge and interest level of the creator/s, along with the caliber of content provided. Some sites have a lot of commercial backing and glitzy features but they seem too cartoonish or dumbed-down for my taste. I’m also instantly turned off by images of red apples and chalkboards, just so you know. I’m OK with advertising, because I know that it takes effort to put forth great content, as long as the information or activities are provided are truly useful, fun, and/or inspiring.

1. My first pick is the now famous Khan Academy site. The creator of the site, Sal Kahn, is both knowledgeable (with three degrees from MIT and one from Harvard) and passionate about helping people learn. His site is a goldmine of free videos demonstrating every possible math concept you can think of, as well as a generous smattering of economics, science, history and SAT prep.

2. Vi Hart Mathmusian’s Youtube Channel: Fibonacci numbers, spirals, fractals, doodles – all about math combined with art.

3. Vi Hart’s personal web site: Besides her Youtube channel, Vi has another site showcasing her math, art, and music related projects.

Vi Hart

Paper mobius strip music box by Vi Hart

4. National Geographic is always an intriguing resource, but they have a few educational projects that look really promising, such as “Population 7 Billion” which involves mapping, human migration, population density and climate change issues. Learning science starts with a reason to learn. Projects like this help make science relevant.

Image from National Geographic

5. The Jason Project is a collaborative effort with The Sea Research Foundation, National Geographic and other organizations to connect students with real scientists and researchers out in the field. There are free downloadable curriculum units on forces & motion, energy, geology, ecology and weather.  There are also digital labs and games to play. My kids and I did this years ago with our homeschool group when the Jason Project team was headed to Antarctica. We did science experiments and other activities related to ice, the ocean, hypothermia, animals, weather, and other Antarctic related topics. It was cool to watch video updates of the research team’s travels and work. The format seems to have changed since then, but it still seems like fun.

JASON Science

6. The Exploratorium is an amazing science museum in San Francisco. My family has visited science museums across the country, but this is our favorite by far. If you are ever in the Bay Area with your kids, this is well worth a visit, and you will want to stay ALL DAY (trust me). But if you can’t make it in person, their website is fun to explore too. There are all sorts of videos, games and activities related to building, sound, colors, geometry, other planets, Polynesian navigation, the ocean, human body, patterns, and general science. All kinds of stuff!


Image from

7. Want more games? Try this one:  This site compiles engineering games from around the web, including bridge design, building roller coasters, space walks, solar car racing, MRI Design, destroying castle walls, and others.

Try Engineering

Image from

8. Want more sleuthing? Try Science Mysteries. Here you’ll find a variety of free mysteries with science-based clues to download and solve, such as “Arctica,” “Strange Dead Bird,” “Poison Dart Frog,” “The Blackout Syndrome,” and “Angry Red Planet.”

Science Mystery

Image from

9. Wondering about STEM career fields? The Science Buddies site has a VERY comprehensive listing of possible careers – some you may have never thought of, like photonics engineer or sustainability specialist. This site is also a great resource for possible science fair projects and topic ideas.

STEM Careers

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10. For older kids and teenagers, I have to include TED on this list. TED, which stands for Technology, Entertainment, and Design, is an ambitious initiative to spread good ideas around the world. Each year the organizers attract scientists, engineers, designers, entrepreneurs, teachers, and other presenters with great ideas to come speak at two sold-out conferences every year. These short presentations are not designed for children, but that is what’s so great about them. Kids will see that these are real people with real ideas that they are working on right now. It’s not has-been science or lecturing. These little videos on everything from “Animations of Unseeable Biology” to “The Magnificence of Spider Silk” to “Distant Time and the Hint of a Multiverse” are what is happening right now and in the future. They are relevant to any kid (or adult) who wonders about the world.  Check it out!


Bonus: Do you have a child interested in computer programming? Here’s a list of recommended sites by my tech-obsessed son:

Code Year – If you know someone who wants to learn programming, here’s a way to start from ground zero.

Stack Overflow – Already know some programming but need help? This is the place to go.

Tutsplus – Lots of tutorials here for learning web development.

Hacker News – For the seriously addicted, a place to find out about the latest happenings in computer technology, etc.


Also, here’s one last website with a list of good open education resources you may not have heard of. Do you have any other favorite sites to share? Please leave a comment below.