As the fall gets into high gear, I will be getting on the road again. If you’re in New York, try to attend NYSCATE this year and register for my session on Singapore Math on Sunday, November 20. If you can come on Saturday, I will be giving a three-hour workshop on NaNoWriMo in the classroom, which will be fun and hands on.

I will also be offering six Singapore Math full-day workshops this fall, starting in October and ending in December. The schedule and links to register for those, and for the conference, are at the bottom of my Professional Development page. If you come, be sure to tell me you saw this website, and you will receive a special little gift!

Tags: NYSCATE, NaNoWriMo, Creative writing, Presentations

Crystal Springs Books recently produced a new software program for number bonds practice. This concept is foundational to Singapore Math. Number bonds can be used for everything from addition and subtraction to understanding fractions, and Singapore Math makes great use of them in its teaching.

The software, intended for grades K-2, is a simple Flash-based program. The CD comes with installers for both Mac OS and Windows. The program is small and snappy, even on older or slower computers; I have tried it on a first-generation Macbook Pro as well as two netbooks. I have used the software with young students and demonstrated it in front of a group of teachers, and this is what we found.

The program comes with four different games, ranging from very basic to more advanced. In the first game, Pond Bonds, children must move frogs to the appropriate lily pad to form correct number bonds. In the second game, Bird Bonds, the purpose is to move the appropriate bird to the right hole in the birdhouse. In this one, each bird is labeled with a number. The third game, Which Number, shows number bonds with one of the numbers missing; students must click on the correct number to complete the bond. The final game, Which Bond, gives students a number at the top of the screen, with two number bonds below. The student must click on the correct question mark where the top number should go.

The games follow the progression of Concrete > Pictorial > Abstract, which is known to lead to student success. Picking up a kicking frog and dropping it on the lily pad, or hearing it splash in the water, triggers concrete sensory feedback, especially when used with a touch screen or interactive white board. Moving birds with numbers on them starts to combine the concrete with the abstract, and the shapes of the holes in the birdhouse mirror the shapes of the number bonds in the next levels. The final two levels use the pictorial and abstract levels to good effect.

The software has several options for customization. For each game, you can set a numerical range, a time limit, and a number of players. Be aware, though, that if you go with the defaults, it may be a recipe for failure; the time limit is set to lowest, and the numbers are set to the highest range, meaning even a very fast adult can’t get a very high score. I wish the defaults had been reversed. On the other hand, if you need to move students quickly through stations, the fast pace can be good. The fastest time may not allow adequate time for learning, though. Once you set the settings for a game, they stay that way until you change them.

One area where this software is lacking is educational feedback for the player. On the early games, if you miss a question, you can go back and try again, but you don’t receive any clues about what went wrong. On the higher games, if you miss one, too bad; you can’t even try again. I would like to see some sort of helpful feedback when mistakes are made.

When you have one player set, the score for the player is displayed at the end of the game. For more players, the others have to sit through each entire set until the scores are displayed at the end. I think more interactive game play would be nice. There is no way to save scores in the software, either. I would recommend that teachers create individual score sheets for students to keep track of their scores and how they improve over time, so they compete against themselves, not against others.

Since the software is Flash-based, it cannot run on iOS devices. I hope they develop a version for that platform soon.

Conclusion: Highly Recommended
Number Bonds is a simple, inexpensive software package that can provide extra addition and subtraction practice in the classroom or at home. Children find it fun and engaging, and it provides good composing and decomposing practice, as well as mental addition and subtraction. I would use it for a wider age range of children; it can be helpful for differentiating, like with more advanced preschoolers and upper elementary students who need foundational number bond practice. It would be nice if the software had a few more features, but I’m sure those features would take away from the software’s speedy response on older hardware. For best results, it should be run on a touch device, so it would be great if it could be installed on iOS or Android in the future.

Pros:
Not expensive
Site license available
Small and fast
Good educational design (concrete > pictorial > abstract)
Fun for children
Range of levels and challenges
Compatible with a wide range of desktops and laptops
Singapore math-based!

Cons:
Not enough feedback on mistakes
Can’t save score data
Settings need to be reset when first played
Not a true multiplayer game
Needs teacher introduction to be most effective (not stand-alone)
Teachers need training to make the most of it, but program-based help is minimal

Tags: Number bonds, Numeracy, Early childhood

This summer I gave a three-hour workshop on Singapore Math model drawing at the NCTM Illuminations Institute in Reston, VA. This was a fun workshop with a great group of people, and we accomplished a lot of model drawing practice and understanding.

I was pleased to see recently that the workshop received a couple of mentions on the web. One is on the thinkfinity site, which is run by Verizon and which I first joined after attending ISTE 2011. The other is from one of the participants, who wrote a blog post mentioning it.

If you are interested in seeing what I can offer your school, please be sure to contact me.

Tags: Singapore Math, Model drawing

ISTE 2011 is up and running, and it’s huge! Look for me there with Conceptua Math at booth 2852 on Wednesday morning before and after our session. I will also be presenting with Arjan Khalsa, the CEO and founder, at this session:
http://www.isteconference.org/ISTE/2011/program/search_results_details.php?sessionid=60744211

If you are a math teacher or homeschooling parent and haven’t seen their free fraction tools yet, please do so at http://conceptuamath.com. The tools are extremely intuitive and valuable, work great with a white board or tablet (but not iPads), and have helped many children understand how fractions work. They are also very compatible with Singapore Math.

Tags: ISTE, Conceptua Math, Fractions

Singapore Math is a rising trend in math education in schools and with homeschoolers, for the simple reason that it works. As an experienced Singapore Math teacher and trainer, I often get the question, “Which Singapore Math series should I use?” This question is posed by both teachers and homeschooling parents, and as more series enter the market, the choice becomes more challenging. In this article, I will present the pros and cons of each current series as I see them. Please feel free to contribute your views in the comments below.

Singapore Math, US Edition, published by Marshall-Cavendish:

This edition has been around the longest in the US. The main difference from the curriculum Singapore was using until recently is that this edition includes some additional problems using US measurements (feet, miles, pounds, etc.). This is the series that Singaporean students used when they scored highest on the TIMSS (international) test.
Pros: 

• Short, focused textbooks and workbooks.
• Clear graphics.
• Emphasis on mental math.
• Clear sequence from one book to the next.
• Follows the best of the Singaporean teaching model.
• Fits the Common Core State Standards well (see this article).
• Decent Homeschooling Guide, from what I hear.

Cons:

• The measurement units don’t follow the Singaporean teaching model; that is, they don’t thoroughly teach one type of measurement before moving on to the next, rather mixing US and metric together. This can cause confusion in students.
• For American teachers, teacher’s manual may be inadequate without further training.
• Doesn’t come with assessments; I used the Practices and Reviews in the textbook for this.
• Needs supplementation with math facts practice.
• Children going from this edition to public school may be missing some subjects, but stronger in others.
• Must be ordered online; shipping is high.

Overall: This is my preferred series despite its shortcomings, which can be easily overcome with a little knowhow and creativity.Buy here: SingaporeMath.com, Inc.

Singapore Math, Standards Edition, published by Marshall-Cavendish:

This edition was created to meet the California learning standards. It is more colorful than the US Edition and covers slightly different topics each year. A comparison chart showing the scope and sequence of the two is available here. This series has been approved by the California State Board of Education.
Pros:

• Designed like the US Edition, with most of the same pros.
• Thorough Teacher’s Guide.
• Comes with Assessments.
• Decent Homeschooling Guide, from what I hear.

Cons: 

• One of the strengths of the Singapore Math curriculum is its focus on mastery of fewer subjects per year. This edition repeats the mistake of many US-designed curricula by putting in too many subjects per year so there is less time for each.
• Must be ordered online; shipping is high.

Overall: Recommended if the child is in California or the state has similar standards to meet.Buy here: SingaporeMath.com, Inc.

Math in Focus, distributed by Houghton Mifflin Harcourt:

This series is new to the US market, and Houghton Mifflin has Americanized it, with the typical large-sized teacher guides, a variety of student books, and manipulatives, packaged in typical school bundles. I have seen the company at trade shows and looked at the materials there, and have requested samples, but none have been forthcoming, so I have not been able to test them out until now. I just discovered the online sampling website they provide, but it’s slow, and I can’t try it out with my students. So while I have been able to see the series to some extent, this review is less in depth than others. Pros:

• Easiest for US public school teachers to adapt to, with explicit guides and scripts.
• Flows better to the middle school/high school Singapore curricula.
• Wide variety of differentiation options.

Cons:

• Expensive.
• Scores in Singapore went down after they implemented this program.
• Less emphasis on mental math.
• Books are larger with more complicated/busier graphics, potentially distracting from the learning process. I like the drawings, but combining them with photos can be confusing.

Overall: Recommended for teachers who don't have prior experience using Singapore Math. For those who do, I need more experience with this program to recommend it or not, but I would stick with the US Edition for now. Homeschooling parents are better off with one of the series from singaporemath.com.Buy here: Greatsource.com

Singapore Math Practice, published by Frank Schaffer Publications:

This series appeared in bookstores a year or two ago, and I made a beeline to it with interest. I put it down almost immediately, though. It appears to capitalize on the popularity of Singapore Math without a thorough understanding of its best principles and practices.
Pros:

• Readily available in bookstores
• May provide extra practice in addition to using one of the other series, but use with caution.

Cons:

• Promotes the use of calculators too early, a big no-no in my book.
• Problems have mistakes and are not well designed.

Buy here: Amazon.com or in bookstores like Barnes & Noble

Tags: Math in Focus, math, math education

MSNBC ran a piece on May 3 about third-grade students learning math using Singapore Math. This report outlines the importance of model drawing for problem solving, and of parent understanding to be on board with it.

The report is well done, except it gives the mistaken impression that the only thing that makes Singapore Math unique is the model drawing approach. I used to think that too, but now I know better; developing number bond-based numeracy is at least as essential, as are other elements of the curriculum.

View the video below:

Visit msnbc.com for breaking news, world news, and news about the economy

Tags: math, math education

I’m so inspired about a new tool to enhance math education. A friend sent me the link to a TED talk (embedded at the bottom of this post) showing the evolution of the Khan Academy into something truly useful for - well, for just about anybody.

I had come across the Khan videos some time ago, and I thought they were useful and well designed to teach more advanced concepts. Since they were not necessarily pertinent to my work, though, I didn’t return to them.

Then I saw this video, and how the Khan Academy has evolved, and I got excited. I set up accounts for several of my tutoring students and asked them to try the site while I checked their previous work, a few minutes of what used to be down time for them. Right there, I have increased the learning efficiency of my tutoring time.

The first student I set up, a fifth-grade girl, got happily into the site right away. When she discovered you could earn badges for different accomplishments, that sold her - and not only that, but she knew that her seven-year-old brother would like the site too.

For teachers, tutors, parents, etc., a wonderful feature is to sign up as a coach and have your student(s) or child(ren) add you as a coach. I think they can even add more than one coach, so both a teacher and a parent coach the same student, for example.

After only using the site for two days, I can already see the progress my students are making, as well as areas in which they are struggling. This will allow me to focus my next session with them better and help them master the skills they need, as well as move more quickly past the ones they have mastered.

The site design is excellent, with only a few minor glitches. Having looked at many educational websites, I can say that this is a rare find. To set up an account, one needs either a Google or a Facebook account. This can be a hurdle in itself; you have to be 18 or older to have a Google account, so for children, they need to either lie about their birth year or use a parent’s account. I did run into this problem in setting up student accounts, unfortunately. Facebook has its own pitfalls; while the minimum age is lower (though still too high for most of my students), parents often have more objections to their children joining that site than Google. Khan Academy recommends teachers signing up for Google Apps for Education; I haven’t looked into that option yet, but I may do so, if I qualify as a private tutor.

Once the signing-up hurdle is jumped, the sign-in process is easy and smooth. The site design is clear and simple to navigate, though I wish the “Add a Coach” link would be easier to find.

The real gem for me is the Practice interface. When you click the Practice link, you face a constellation of skills, with Addition 1 at the top. As you demonstrate proficiency, you earn a star in that constellation, and the graphic indicates the suggested skills to work on next.

The interface is simple but effective. When you start to practice, the problems show up as images, and you enter the correct answer in a text box. What’s great about it is that it’s Flash-free, meaning it works on iPhones, iPads, etc., making fun math practice freely, and widely, available.

There was another minor technical glitch, though. At first, I was using my little Asus netbook tablet, and I was thrilled to discover the “Show scratch pad” link. This enables a vertical bar on the left of the screen showing tools like a pencil, eraser, etc. that allow you to write on the screen to do your work, like on a note pad. On the tablet, this was awesome, because my students could use the stylus like a pen to work out their answers right on the screen. I thought the iPad would be as good or better for this, but instead, the touch interface interacts only with the browser controls (like scrolling up and down), and I couldn’t make it register any marks. I’m not sure if this is browser-related, a site programming problem, or an issue in the iOS. It would be great if this could be solved. But it would be ideal for a teacher with an interactive whiteboard as well.

To test the system further, I chose a math topic about which I am very rusty. When I clicked on the subject’s button, I saw a problem that stumped me completely. What to do?

In a classroom situation, a shy student might just sit there and be miserable. But in this tool, right below my choices were the friendly words, “Need help?” and a selection of videos that could show me what I needed to know to succeed in this topic. Better still, I didn’t lose any points by watching the videos - though I would if I asked for a hint.

Are the videos perfect? No, but they’re good, and they have the advantage of being easy to watch over and over until you understand the concept. It would be great if he used more of the methodology in Singapore Math to teach the basic concepts, but we can’t have everything at once. Maybe someday.

One criticism of Singapore Math I have heard is that it needs more skills practice. I think this site is one way for a student to get this practice in a low-key, interactive, fun way. It’s also a terrific tool for students and teachers to improve their learning progress, and for anyone who wants to learn.

By the way, math isn’t the only subject addressed on that site, though I think it’s the first and probably the most thoroughly done. The other subjects, including test prep, are worth visiting too.

Tags: math, Technology, TED Talks, Algebra

And now for something completely different! When I am not exploring math education or writing fiction, I love to make things. This includes everything from cooking to needle felting to making jewelry. So when instructables.com opened their Pi Day pie contest, I had to enter.

Creating my entry required a great deal of calculation, from halving or quartering recipes to guesstimating how long it would take to bake this unique cookie, cake, and ice cream pie. Fortunately, I’ve had lots of experience with estimation, and I couldn’t have hoped for this pie to turn out better. My friends who shared it loved it too.

If you’d like to see the pie and hopefully vote for it, please head on over to this page. There are also lots of other fabulous entries, and you can vote as often as you’d like. Enjoy!

Tags: Pi Day, baking

Yesterday was March 14, or 3/14, or Pi Day. It’s a great day to celebrate the circle, and that most extraordinary number, pi.

With my second grade math club, I did several activities on my new teacher download, Pi Day Activities. These included cutting a circle, measuring a circle, and eating pie. We didn’t have enough time to play Pi Tag, though I’m sure we’ll be doing that one week soon!

I also created a poster showing almost 1,500 digits of pie. You can download it for free either from my site or from my TeachersPayTeachers.com link.

Happy Pi Day!

Tags: math, Pi, Pi Day, Circles

A Delaware school district has successfully implemented Singapore Math, raising enjoyment, understanding, and test scores. This article describes their success. Here is one example:

Mount Pleasant Elementary Principal Joyce Skrobot did not need to be convinced to add Singapore math to the curriculum. Her school piloted the program over the past four years in some second-grade classes, and, on state tests, they outperformed the classes that did not use the math, she said.

"It really establishes a strong foundation of math skills with a lot of repetition," she said. "It's a very concrete approach to teaching."

The district plans to offer parent workshops to explain the differences in the Singapore approach, a key component of long-term success.

Tags: Delaware, math, math education

A video demonstrating how ten frames can be used to develop number sense was posted at http://www.youtube.com/watch?v=zQxS5Z3UHKk&feature=player_embedded. (They disabled embedded on external sites, so you will have to click to see it.)

The video shows progression from counting-on with touching, or the concrete stage, to the pictorial stage of being able to look at ten frames and see how many dots are present. Early in the video, it says the child is a kinesthetic learner, which may be true, but touching the objects is a natural early stage for anyone. So touching the objects doesn’t necessarily mean the child is a kinesthetic learner, but they may be at the concrete stage of learning a certain concept.

The clip does a nice job of showing how a teacher can help a student one-on-one (though I would have liked to see the teacher doing more guiding and less instructing), but what about teaching larger groups of children? There are always issues of permission when dealing with groups; however, I think it would help teachers if they could see how to use this in a larger setting. This is something I can model when offering professional development at a school visit.

Tags: Ten-frames, Concrete, Manipulatives, math

If you’re going to tell triangle jokes, you should do them in threes, right? So here’s the third one, also original:

Q: Which triangles are the best conversationalists?

A: The acute ones. The others are either too obtuse or always right.

(That one got a laugh today!)

Tags: Humor

I came across this joke tonight in a blog comment.

Q: What did the triangle say to the circle?

A: Your life seems so pointless.


And a bonus original joke that I just made up:

Q: Which triangles are the most likely to get the point?

A: The acute ones. The others ones are just too obtuse.

Let me know if you thought it was funny!

Tags: jokes

In August 2010, Achieve.org produced a report comparing the Common Core State Standards with the Singapore Math syllabus. I found the report interesting, as it showed that there are many similarities between these standards and Singapore’s syllabus, though in some ways, the CCSS document is clearer in its expectations. Also, Singapore uses the British system of O-level and A-level achievement. Their O-level high school curriculum is slightly less rigorous than ours, but their A-level curriculum is more rigorous than our standard high school curriculum.

I drew the conclusion from reading the report that adopting Singapore Math could be a positive step towards aligning to the CCSS.

Achieve is an independent non-profit dedicated to raising academic standards in the US. You can read the full report below.
Comparing the Common Core State Standards and Singapore’s Mathematics Syllabus

Tags: Standards, math education

A great article titled Waiting for Supermath came through my inbox today. It includes commentary on a video (below) of a third grader showing how she solves a four-digit addition problem using what she learns at school, or the Investigations curriculum, versus what her mother (a math intervention specialist) teaches at home, the traditional “stacking” algorithm.

What strikes me most about the video is that the first method, using the graphic model, shows what seems to me an overuse of the conceptual level of addition.

One strength of Singapore Math is that it starts with the conceptual level, which is essential, but then it moves to the abstract. In this process, the student starts with concrete representations of a problem, like manipulatives, then to pictorial or graphic representations, and finally to the algorithm, once they have mastered the concept.

But in the video, the girl starts out solving the problem with what could be drawings of base 10 blocks - and way too many of them. This is keeping her stuck at the concrete stage and leads to inefficiency and inaccuracy in her calculations.

It also strikes me, as the video points out in the end, that this method of teaching creates the myth that larger numbers are harder to calculate. Is this what we want to perpetuate in our students? I know if I had, I wouldn’t have had a group of second and third graders who decided, on their own, to learn 50 or more digits of pi.

One other note: I did use Investigations for one year in a middle school classroom. That was the year that some parents and I convinced the administration to finally adopt a curriculum that made sense. And what did they choose? Singapore Math!

Watch the video:

Tags: Constructivism

President Obama would like to know how South Korea has risen up to have one of the fastest-growing economies and best-educated workers in just over a generation. Rather than look to a magic fix, The Lost Seoul addresses some cultural differences between South Korea and the US in this blog post. One important difference he mentions is attitude. If you ask an American student if he or she is good at math, you will usually get a straightforward answer. If you ask a South Korean student the same thing, he or she won’t know how to answer. The question doesn’t compute.

The Lost Seoul suggests that the reason for this is because in the US, we equate math ability with genetic tendency - you inherit it from your parents - which is self-limiting for those who have parents who don’t believe they are good in math. And if they don’t think they are good in math, Americans won’t pursue it past high school. But in South Korea, math is just something they do, probably more like reading in the US. Adults in the US don’t stop reading after high school just because they might not have been the best or fastest at it in school. It’s part of life, in everything from sports or fashion magazines to professional journals. I found the post interesting and informative, and I recommend checking it out.

Tags: Korea, math education

Today in my Math Mavens program, we opened the book You Can Count on Monsters by Richard Evan Schwartz for the first time. This is a book I bought because I heard glowing reviews of it on NPR.

The concept of the book is teaching prime and composite numbers through colorful, geometrical monsters. It is written for any age, from preschool on up, and my students really appreciated it. They had a lot of fun looking at the monsters, spotting the prime monsters hidden inside the composite monsters, and describing what they saw. For example, one said the 20 monster looked like “two innocent two-monsters held in custody by evil nacho chips.”

For fans of Singapore Math or number bonds in general, you will also appreciate how each number is represented with a number of dots, the numeral, and a multiplication number bond for composite numbers. All in all, it makes a powerful set of connections for students between numbers, images and fun.

The book covers numbers 1 through 100, with an introductory section that explains factoring, prime and composite numbers, and how the book is designed, all with colorful images and not too wordy. A section in the back has a further exploration of prime numbers. A wonderful enrichment for any math education!

To see inside or order the book on Amazon, click below:

Scarsdale, NY is a model district in terms of scores and success. They attribute this success to five building blocks in their curriculum: Singapore Math, inquiry approaches to science and social studies, fluency in information technology, and creative arts. Yet they are having to cut teachers and programs due to budget constraints. They contrast this to China, which funded five educators to visit their district.

Will Scarsdale have to cut back on their successful programs? Meetings of the Scarsdale Forum are happening during February. Read more at this Patch article.

Tags: Scarsdale, Singapore Math

We hear plenty of talk about teaching and reinforcing basic skills in math. Yes, these are very important, but computation skills aren’t what lead to breakthroughs and new discoveries; new ways of thinking do, right?

This young woman exemplifies real creativity in mathematical thinking. I find this so inspiring. Investigating mathematical principles through art: what a concept!

Tags: doodling, math education

Singapore Math summer programs come to Westchester, NY! Do you have children who would benefit from a summer experience learning math in the proven Singapore Math way? Send them to the brand new program offered this summer by experienced Singapore Math teacher, instructor and trainer Susan Midlarsky. Not only will they learn a lot, but it will be fun! Find out more at singaporemathny.com. Register early - space is limited!

Tags: Singapore Math strategies, Singapore Math camp

An article published by the Rodel Foundation of Delaware describes how Kuumba Academy took a serious approach to remediating the problem of poor math achievement. They adopted a Singapore Math curriculum, and with it, they gave their teachers “intensive, on-going professional development to deepen teachers’ understanding of math instruction at the elementary level.” The school also implemented parent workshops and a “Bring Your Parent to School Day,” which would help parents and guardians understand the sometimes very different approaches Singapore Math takes.

One minor incorrect point the article states is that Singapore Math uses math sprints to strengthen math skills. Sprints were developed by Yoram Sagher, a US professor, to supplement math fact practice in the Singapore Math curriculum. Using them can help, although they are not the only effective math skills practice approach.

Adopting the Singapore Math curriculum, along with training teachers well and using sprints, led to a complete turnaround in the school’s math test results. As the article states:

Since Kuumba began its partnership with the Vision Network, the school has seen phenomenal growth in math scores. Not only are students no longer falling behind, they are exceeding state performance in many grades in math. In just 3 years, Kuumba went from 49% of students meeting the standard school-wide to 87% proficient, as measured by the DSTP.

Tags: Kuumba academy, Singapore Math strategies, Professional Development

It snowed today - a lot - canceling all my plans and making it a perfect day to get things done at home. So I created the short movie below. I hope you enjoy it as much as I enjoyed making it! This one was taken - and highly modified - from a joke told in Math Jokes 4 Mathy Folks.

Tags: Humor, jokes, math

How do you explain the concept behind the formula for the lateral surface area of a cylinder, which is 2Πrh? I ran into this question when tutoring a student to prepare for the New York State Integrated Algebra Regents exam. (For some reason, this exam contains a lot of geometry.) The lateral surface area is the area of the cylinder’s surface that does not include the circular ends.

If you look at the cylinder, it resembles a can. If you imagine it is a can of something, the lateral surface area is what the label covers.

So to show the concept behind the formula, we took a can of organic garbanzo beans out of the kitchen cabinet. Fortunately, it had a label that was easy to peel off.

First we measured the width of the label. Next we measured the diameter of the can (the 2r, or twice the radius), and multiplied it by Π. Comparing the two widths, the rectangular label width pretty much matched the formula for the circumference, or a little over three times the diameter exactly! And since the label is a rectangle, to get the area, we multiplied the length by the height.

So we discovered, by this exploration, that the width of the label is equal to the circumference of the circular top. Therefore the formula made sense to the student, and we had fun making it concrete. If she doesn’t remember the formula on the test, I’m sure she’ll be able to access the concept to recreate the formula at the point - and that will demonstrate true understanding.

Tags: Geometry, Formula, Concepts

The author of Social Media for Trainers and I have been having an interesting debate about the place of higher math education in schools. I had read her book and found it useful, so I looked up her Facebook page. There, under the heading, “Stop teaching math,” she placed a link on her Facebook page to the blog article titled “The Case Against Math.” Of course I found this provocative and clicked over to read the article.

While I agree with the thesis of the article - that the way we teach math and value it as a proxy for measuring intelligence is not useful, and that it should be changed - I do not think we should reduce or eliminate it as a requirement in education. Instead, I agree for the most part with how the article’s author puts it:

“If we must teach math, teach it  as if math was just one aspect of the larger concepts and questions that are the main thrust of education: critical thinking, problem solving, communication, empathy, and creativity. If we must teach math, teach it through music, art, science, technology, history, cooking, construction, engineering etc. because math as an abstract system is useful to very few of our students. If we must teach math, focus less on the answers and the algorithms for specific types of problems and focus more on the questions and the processes of problems.”

I do think that teaching math in an integrated way is best, but I also see merit in teaching math as a subject unto itself, as long as it’s taught in ways that make sense. The process of teaching through problem solving and from conceptual to abstract allows math to make sense to all students I’ve encountered, and problem solving therefore becomes a fun challenge, not a chore.

As I mentioned on the Facebook page, I once had a friend who was working as a carpenter. He asked for my help in figuring out how long a piece of wood needed to be to complete an attic renovation project. I showed him how to solve the problem using the Pythagorean theorem. This was before I became a teacher, but he told me that if he had had teachers like me in high school, he probably would not have left school, as this was useful stuff to know.

The author’s response was to ask 3,000 Twitter followers for examples of using advanced math in their everyday lives. She received one tweet about a problem similar to the carpentry one, and one about helping a child with trigonometry homework.

This doesn’t surprise me if the vast majority of her followers are Americans. I would love to know, though, if we would get a different response from people raised in other countries, especially those in countries that have consistently scored highly in math. If no studies have been done on this, I would like to study it myself. Does how we are raised to think of math affect how we use it (or don’t) in our daily lives, or is the subject objectively useless to all but scientists and engineers and taught only as a carry-over from ancient times? What do you think?

UPDATE: I discussed this topic today with a student of mine who is “unschooled” and started fifth-grade Singapore math with me when she was 15 years old. Two and a half years later, she is at high school Algebra level. Her main interest is fashion design, and she’s been attending high school fashion design classes for a couple of years. She told me that she was pleased to put her fraction knowledge to use in her sewing class last spring. That’s only one story; do you have your own?

Tags: Higher education, math

For my educator friends and colleagues, I have added a new multiplication chart lesson plan, complete with reproducible handouts, to TeachersPayTeachers.com. It is free to download and use. It can be used in a classroom, in a homeschooling setting, or in a special needs or remedial context.

The lesson is aligned to the Common Core Standards and includes objectives, materials, and descriptions of procedures, follow-ups and adaptations. Please let me know if you find it useful, and if you do, please add a rating to the TeachersPayTeachers site.

Download the lesson plan here.

Tags: Multiplication tables

The more I travel and meet people, the more I find that most adults in the US have difficulty with math. I read a comment by a woman from Eastern Europe who found that while she was a mediocre math student in her home country, she was miles ahead of American students when she moved here. She couldn’t understand why, with all the time and finance poured into math education here, including an average of 1.5 hours per day of math class, her children were progressing in math far less well than she had when growing up.

I think part of the reason is that we have a couple of generations of adults who just don’t have a strong grasp of math concepts, especially when it comes to basic number sense. Various adults have approached me and asked if I would teach a math class so at least they wouldn’t pass on their own math phobias to their children, and maybe they could even help their children with their homework and learning. The latest of these I met were a couple of lovely older women in Oklahoma who were staying at the same hotel as I for an agriculture convention.

Reaching these adults presents a challenge because of the distance. Attending a teacher workshop would be overkill and too expensive. So I came up with the idea, what about an online course offering math fundamentals for adults? I think it could benefit a lot of people.

What do you think? Do you, or anyone you know, think you or they might benefit from it? If you were to take such a course, what would you want to be part of it? Let me know!

About two weeks ago, a post titled “Singapore Math Is ‘Our Dirty Little Secret’” appeared on the Core Knowledge blog. It criticized the New York Times article about Singapore Math that appeared on October 1. Apparently, the author believes that the poor state of math education in the US is due to what he calls “reform math.” This ignores an entire generation of math-phobic adults who learned math through “traditional” methods, and most likely instigated the reform movement due to their dissatisfaction with those methods.

While the curricula based purely on constructivist approaches have their limitations, the idea that Singapore Math is a traditional approach is mistaken. It’s better than traditional approaches.

Below are the comments I wrote on the blog:

As a long-time Singapore Math educator and trainer, I have to disagree with a few points in this post. Overall, it seems to be advocating a "traditional" approach to math, the same approach that has led to poor US performance in math and science in the last few decades and an epidemic of math phobia among American adults. This "traditional" approach has also led to one of the main reasons elementary math education suffers these days: too many educators had poor math education and don't understand the concepts themselves, so they have no idea how to teach it to the children. They are afraid of the subject, so how can they be successful in teaching their students? If they were taught algorithms with no idea of the workings behind them, they cannot pass an understanding of the workings on to their students.

When I teach my workshops, one of the things I see is when I demonstrate one of the basic four operations on whole numbers - addition, subtraction, multiplication and division - with number disks on a place value chart, many of the participants have an "Aha!" moment. So that's how it works, they realize. And once they have this understanding and practice it, teaching it to the students - and being able to be flexible enough in their approaches to reach all students - becomes a reachable goal.

This use of place value disks is an example of the concrete stage of concrete > pictorial > abstract that Singapore Math is based upon. The textbooks are full of diagrams that show the place value chart being used in this way, but those diagrams are meant to illustrate what the students have already done with the place value charts and disks, which then builds into understanding of the algorithm and how it works. And yes, this is part of the process of learning from conceptual understanding to algorithm built into the curriculum. Manipulatives can be very powerful, and I find them necessary for most students. There are always the few who will understand no matter what, but those are not the students we need to help.

I had used the textbooks and workbooks for a few years, even with training, without understanding this pedagogy, and was somewhat successful - just because I understand math myself. But when I became equipped with the deeper understandings mentioned above, I became a much better math teacher, able to differentiate and address different learning needs.

Regarding the model-drawing books, the cynical comment about them in this post is misplaced. Some teachers may use the steps for model drawing as a rote formula, but that's not how they are intended. If you have never learned how to do model drawing, you need some kind of instruction. Then after that, the steps are just there to remind you until they are internalized and personalized.

I have taught several model-drawing workshops in which participants (mostly high school teachers) have said the most valuable part of the workshop for them was the step, "Write your answer statement first." This is a sentence with a blank for the answer, reworded from the question in the problem. It serves the purpose of refocusing the student at the end of the problem when they need to find which of the many calculations they may have worked is actually the answer to the problem. The Singapore workbook problems are set up this way, but without instruction, children may miss out on this step. I know I did!

I agree that the purely constructivist math approaches leave a lot to be desired, but the idea that Singapore Math has no constructivist elements is incorrect. I think that if it is taught well, it strikes a good balance between constructivist and elementary knowledge in such a way that children can master the math knowledge they need to succeed – and I have seen this success in my own students over the years.

Tags: blog, Numeracy

My first trip to NCTM is over, and I’m glad I went. Although the setup had a few glitches, like an LCD projector that didn’t want to project from my laptop, my presentation on problem solving using model drawing went well, with close to 180 participants. Many of them came back to the booth, interested in further learning, and some bought books and materials or inquired about future opportunities to develop this skill. I’m really pleased about this, because it means more children may be better equipped to enjoy and understand math.

The booth was busy the whole day, and I demonstrated model drawing with word problems a number of times. That was fun and always drew attention. It’s great to see that “Aha!” moment when teachers see what a powerful tool model drawing is to visualize a word problem. I even used model drawing today in a tutoring session with a high school student who was studying for the PSAT. We were going over some of the problems about which she had questions, and I showed her how to model a problem involving ratios. Using the model drawing method made the answer visually obvious.

It was also great to spend time with team members and colleagues, as well as to meet new people. I hope some of the new contacts will develop into lasting relationships.

If you were a participant in my workshop, I do plan to post the answer key to the model drawing questions here shortly. Check back; they should be here by Monday. Thank you!

Tags: NCTM, math, math education

After a long day of arriving and helping to set up the SDE booth, I had a little time to look around the NCTM bookstore. (NCTM, in case you don’t know, is the National Council of Teachers of Mathematics and the host of this conference.) There were some interesting books, but the one I just HAD to buy was Math Jokes 4 Mathy Folks by G. Patrick Vennebush. How could I resist? It shows what a math geek I am that I was laughing out loud while reading some of the jokes. This will be a great resource for any of the math presentations and/or classes I give.

While I don’t feel good about reproducing any of the contents of the book on my blog, I can share a joke told by a wonderful woman I met tonight.

Q: What did the zero say to the eight?
A: Nice belt!

Check the book out here:

Besides doing the presentation tomorrow, I am really looking forward to meeting Greg Tang, my favorite author of math-themed picture books. He will be stopping by the SDE booth (#614) at 11:30 tomorrow. If I’d known before I left home, I’d have brought my copies of his books to be autographed!

Tags: jokes, Funny, math

Are you looking for ideas about how to engage students in math, or show them how it applies to the real world? Here is a fun one for sports lovers. John Roach at msnbc.com recently published an article called “The Math and Science of Baseball.” It outlines various ways in which math and science have been applied to the sport.

We all know about batting averages, but did you know scientists have analyzed everything from how likely it is that the best team will win with the current number of games vs. the ideal number of games per season, that mathematical models judging fielding ability have been created, and that statisticians have studied managerial style in relation to different types of teams?

Also, do you know which is faster, a head-first or a feet-first slide into base? Check out the article to find out - and maybe include some of these fascinating facts in your next math class!

Tags: Science, math, baseball

Teachers of grades 3-5, are you interested in a grant for professional development in science and math? Parents and students of a 3-5 grade teacher, are you interested in helping your teacher have a great opportunity to learn more math and science? The Mickelson Exxon-Mobile Teachers Academy is accepting applications through October 31. Be sure to apply soon!

Tags: grants, math education, Teachers

Imagine trying to pay for a doughnut and not knowing if a $10 bill is enough.

Imagine not knowing which is more, 5 or 4.

Imagine never having a sense of time, so you are always early or late for things. Or someone gives you an hour to complete a task, and you have no idea how long that is or how to pace yourself.

Imagine never being able to retain the difference between left and right.

Imagine being in high school and understanding the concepts of algebra, but being unable to do basic addition and subtraction, let alone the higher operations.

Imagine being gifted in many, many academic areas, but having such difficulty in these areas that you “average out” so your school system never qualifies you for either the gifted programs or the special needs support you so badly need.

Imagine taking a summer job cleaning hotel rooms and being repeatedly reprimanded because you can’t keep all the steps in your head, forgetting the towels one time, the soaps another, dusting the counters yet another time.

Imagine that most of your teachers don’t understand, say thoughtless and clueless things about your disability, and some even try to block you from getting the special services and supports that you need.

Imagine the stress and anxiety that comes from not understanding what is wrong with you, why you can’t get the simplest things that come to all your peers so easily.

If you take the time to imagine all this, you might get a glimpse of the feelings conveyed by Samantha Abeel in her memoir, My Thirteenth Winter. I just finished rereading it, because I wanted to remind myself about what it is like for a person with dyscalculia.

This disability, unlike dyslexia, is one that I had never been trained in during my teacher education, though current research suggests in might be related to dyslexia. I think it is one of the lesser-understood disabilities, at least by the general population. Reading this memoir helped me become a better and more sensitive teacher, but it also raises some questions: how best to support people with this disability? After all, it makes living a regular life very challenging, between having to handle money (a big one) to using directions to get somewhere (though GPS can help) to being able to manage your time.

I have thought about Singapore Math in relation to this. In a session with Dr. Yeap Ban Har this summer, he mentioned that being able to do mental math and compute with number sense should be able to be done by all students without calculators, but that those with disabilities who can understand the concepts but cannot compute should be able to use calculators.

Part of the answer might be to look at multiple intelligence theory and use the student’s own strengths. Samantha Abeel is very strong in her visual and literary abilities. For her, a program that teaches math facts through poems, stories, and/or pictures might have been helpful, as it would use different brain pathways to help her retain these facts. One promising resource is here; please share others in the comments if you have any suggestions. I also had some success making multiplication table music CDs with a couple of math classes, and they seemed to help my students with learning differences the most.

Another interesting project with some research to back it up is a software program for young children that incorporates a game to teach basic number sense. It is a Java-based game, so it starts up slowly on my computer, and it’s not super-polished or professional looking, but the pedagogy looks solid. The game is called The Number Race. The full published research article about its effectiveness is at this site. In summary, the students who used it had some improvement in kindergarten, but it did not hold over time. A one-shot deal is not good enough; they need repeated help and practice.

If you would like to add anything about dyscalculia and how to help students who have it, please do so in the comments below. Thanks for reading.

Tags: Special needs, learning differences, learning disability, dyslexia, dyscalculia, math software, compensation