This post was originally published on the Patch on August 5, 2011.
I was recently working with a young student who had a hard time figuring out when to add commas or periods in his writing. I had given him a worksheet made from a paragraph I wrote and from which I removed proper capitalization and end punctuation. All he had to do was rewrite the paragraph with correct periods and capitals.

Even though this sounds simple, he had a hard time determining where a period should go. Instead, he sometimes added a comma instead or skipped a period entirely.

After trying several approaches to help him, a new thought occurred to me. “Listen to the music of the words,” I said. I compared punctuation to rests in musical notation, with which he has some experience. “Commas are like short rests, and periods are like long rests.”

I then hummed the paragraph without the words for him once, and then again while pointing to the words I was humming so he could follow along. He said it was almost like he could hear me saying the sentences. I said, “Yes, the music and rhythm of language help give it meaning.” To show some contrast, I read the same paragraph in an absolute monotone with no pauses for punctuation whatsoever. “Much more boring, isn’t it?” I said, and he agreed.

With this new tool under his belt, my student was able to successfully detect when to add periods in the rest of the paragraph. He continues to use this tool months later.

I have taught this method to other struggling students, and it’s helped them, too. A search for similar methods didn’t turn up anything online, so I wonder if this is a new idea. I hope this way to use “musical intelligence” adds another useful tool to other writing teachers’ tool kits!

Tags: Multiple intelligences, Music, Proofreading

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

As of today, I am live as a blogger on the local Patch, an online newspaper! Read my first entry concerning an educational green building possibility here: http://peekskill.patch.com/blog_posts/verplanck-enhancement-plan-and-education .

Tags: blog, green building

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

Like so many others, I am compelled to plant seeds for a garden when spring scents warm the air. This year I will grow a variety of vegetables in a raised bed I plan to construct out of mostly found materials from a nearby beach.

To start the process, I planted some seeds in an eggshell planter. I saved eggshells from breakfast eggs, poked holes in the bottoms, filled them with soil, added seeds, and put them in the egg carton to sprout.

In just a few days, a whole group of seeds turned into fast-growing seedlings that had to be transplanted as soon as possible. These are now growing in their temporary planter made from a bin found on the beach.

Other seeds are just now springing up, and yet other eggshells look quite devoid of life for now. The different plants have different timings. Am I worried? Do I fret? Of course not, and if I did, my seed packages would comfort me with messages like, “Days to germinate: 12-28 days.” It’s fine, it’s normal, it’s nothing to worry about. Some seeds are little race cars to the sun, while others are more relaxed, laid back, and “zen” about the whole thing.

But what about when it comes to our children? In his recent Differentiation Strategies workshop, Jim Grant said something that stuck in my mind. He said that each child learns to crawl at her own pace, to speak, to walk, to draw, to dance, and everything else. There are developmental milestones, but no fixed criteria. Yet when it comes to school tie, our “standard” for admission is simply five birthday candles on the birthday cake, with very little or no flexibility in timing.

No wonder standardized testing has so many problems. It’s like taking a bunch of seeds from different vegetables and trying to sow and harvest them at exactly the same time. Some will be perfectly ripe, but most will be just flowering or else rotting on the vine.

A good school system would be like a good gardener: able to treat the different children like the different seeds they are, planting them in the correct educational environment (soil) at the right age (season) for that person, and giving them the knowledge and experiences (fertilizer) they need to grow. What we would lose in institutional efficiency in admitting students, we would gain in collective intelligence, confidence and empowerment. Can it happen?

Tags: gardening, Standards, Seeds

It’s been quite a week for me to be immersed in school reform. After speaking at the Voyager’s Community School education conference, I heard John Taylor Gatto and Ron Miller speak, among other reformers. I also went to an excellent differentiated instruction workshop with Jim Grant last Tuesday, in which he laid out many of the problems facing education today. On top of that, I just watched the film Waiting for Superman.

Brewing on all of this led me to this question:

What is education in our public schools for?

Ask different people and you’ll get different responses.

Is it for manufacturing criminals? (If not, then why are certain schools so good at this?)

Is it to break up our family units and create artificial divisions within our society? (Some would say yes.)

Is it to prepare students for life in the real world? (Many would claim this is its purpose, but then why is it missing so many essential elements, like how to clean your house, manage credit, deal with relationships, be a good parent, and more?)

Is it to foster creativity in our students? (If it were, then testing would go away and the money would go into currently disposable arts programs.)

Is it to provide guaranteed jobs for teachers, many of whom don’t care anymore - or as one reformer put it, they decided to retire two years ago but didn’t bother to inform the principal? (Some would say this is a big part and problem facing public education.)

Is it to prepare students for college? (If so, then it is failing there too. But what is the purpose of college?)

It is so easy to get lost in the morass of problems facing education today. Instead, I choose to take a step back and ask a new question:

What should the purpose of education be?

Many would say it is to prepare students for college, to get them a good job and success in life. This is partly true. Students who wouldn’t normally have a chance at college see going to college as a huge step towards success. This is the case especially now, because more and more, in order to have a decent job in the US, you must have a college degree. But is that enough?

For those who were fortunate enough to achieve going to college fairly easily, many would say it is not enough. In fact, the approach to education before and during college might even limit our thinking too much.

Instead, what about this: the job of education should be to provide what the students need in order to grow, and to not limit what they are able to achieve.

Imagine a United States in which this was the norm, in which everyone is able to reach their fullest potential (whatever that might be) and has the confidence and security to not have to compete and compare themselves with others.

I was lucky to teach in a school with this philosophy and the know-how to achieve it, The Garden Road School. What a difference that approach makes.

Tags: John Taylor Gatto, Waiting for Superman

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

I recently watched a video of a teacher helping a student master tens and ones using ten frames and unifix cubes. While the video showed some of the ways Singapore Math teaches number sense well, a few things about the teaching style struck me. These are pertinent to tutoring because it was a one-on-one situation.
 
One thing that stood out was that the teacher sometimes gave the answer to the student before the student had a chance to think. This is a mistake that is so easy to make; the tutor knows the answer, and the child doesn't, so why not tell the student what the answer is? That will help them learn, right?

The thing is, any good teacher or tutor knows that the best learning happens when the child discovers something for him or herself. For most people, if they know the answer, it’s hard to hold themselves back from giving it to the child. But the best teachers guide or lead children to making their own discoveries. If children are moving in the wrong direction, a good tutor guides them toward successful results. This requires a knowledge of why students are making the mistake they are making.

For example, say I’m checking some pre-algebra homework. The problem the student has to solve is 3x + 4 = 73. The student’s answer is x = 25-2/3. From my experience working with students, I will know that the student performed all steps correctly except for adding 4 to 73 instead of subtracting it. Depending on the pattern that shows up with other problems, it could be a fluke, or it could point to the need to review the concept of positive and negative integers, or how to solve an equation with a variable. If it’s the latter, we may need to go back to the concrete stage of learning and work our way back up to the abstract (equation) level. Throughout the process, I will not give the child the answers, but will ask them to justify their answers each step of the way. That way, they are more likely to catch their own mistakes and correct faulty thinking.

Being a good tutor requires more than just mastery of a subject. It requires an understanding of how the subject works, how students learn, what kinds of mistakes they might make and what those mean in terms of review. Most of all, it means being able to guide students toward their own voyages of discovery and learning.

Tags: Tutor

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

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

One of my favorite humor bloggers is Allie Brosh, author of Hyperbole and a Half. I’ve been catching up on reading her posts lately, and this one caught my eye tonight: Hyperbole and a Half: Long Division Isn't Real. (If you visit the link, just be forewarned that she uses the f-word once in her post.)

This is how she describes her mom’s attempt to teach her long division in fourth grade, the year Allie was homeschooled. (Her actual post contains an awesome drawing about it too, so visit it if you can):

My mom was like "First, you draw a line with a little hang-y tail!  Then you write the big number inside the little half-box.  Then you write the little number on the outside!  Now, divide the the little number into the littlest part of the big number that is at least as big as the little number.  It probably won't fit exactly, but that's okay.  Figure out how many times it fits all the way and write that number on top of the box.  Now, write the number that the little number does fit into underneath the number that it doesn't fit into and subtract them.  Then draw a line.  Then write your answer under the line.  Then bring the next number in the big number down next to the number you just wrote.  Then hop on one foot and punch yourself in the face while singing Twinkle, Twinkle Little Star... "

Does that sound familiar?

That’s the pitfall of trying to teach the “how” of long division before the student understands the concept.

Teaching students division the Singapore way - by starting with place value disks and understanding what division is, working with the concept, and gradually connecting it with the algorithm, along with learning alternative ways of dividing - has been a life (mind?) saver for my students learning to divide.

Do you have a story about long division?

Tags: Humor

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

Among my students are three brothers in middle school whom I tutor in writing. They are all honors students whose parents hired me as a tutor for enrichment.

One activity I’m doing with them is to write a letter to the president. It’s simple enough to do: the White House website has an easy-to-use contact form, like those found on many websites. Or, of course, the letter can be mailed.

The activity sounds straightforward: the students should write about an issue that is important to them and send the letter to the president. It is an opportunity to discuss civics and current events, and to practice formal letter writing.

But it’s harder than it sounds, even with advanced students in middle school. First, despite their school classes in current events, most American children don’t feel connected to what’s happening in the world or in politics, or why these issues should matter to them. So we have spent a lot of time discussing some hot topics, like health care, food supplies, international wars and climate change. They really didn’t know much about any of these areas, so it was an opportunity for them to research and get to know more.

Once they each picked a topic, the next hurdle was what to say and how to say it. Getting the letters factually correct and written in a respectful tone was the first hurdle. Making sure the issues really mattered to them was the next. Since two of them chose health care costs as a topic, we spent a while discussing what health care would cost when they become adults if the costs keep rising the way they are now. We discussed the other costs of living and what they would be able to afford if they had to pay, say, $20,000 or more per year in health care costs. This helped bring the issue home to them.

This week I also encouraged them to put themselves inside the picture: let the president know why these issues matter to them. I think if more people involved themselves in this way, either as children or as adults, it could change the quality of our country for the better.

Oh, and as I was leaving today, their mom told me, with a beaming smile, that the boys really enjoy our classes. Imagine, middle school boys who enjoy having an extra hour and a half of school-like work added to their schedule each week!

Tags: Social studies, Current events, Politics

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

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

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

In a training today, I learned about sentence frames. These are helpful for English language learners, and also native speakers, to develop understanding of math concepts by filling in a statement with blanks in it.

In researching this further, I came across a wiki that contains a number of sentence frames for various California math standards categorized by grade level. These can be used in any context to fit your math teaching. You can even contribute your own sentences by joining the wiki.

Go to the wiki here: http://mathsentenceframes.wikispaces.com/

Tags: wiki, Singapore Math strategies, English language learners

Teachers in my math workshops like me to share some multiplication and factoring tips I teach my students. These help with number sense as well. I hope they can be useful for you too.

Tips for Multiplying Whole Numbers

• Times 2: Double the number. If multiplying by 2, the result will always be even.
• Times 3: Triple the number. Products alternate odd and even (3, 6, 9, 12, etc.).
• Times 4: Double the number twice. The result will always be even.
• Times 5: The result must have a 5 or a 0 in the ones place.
• Times 6: Triple the number and double it, in whichever order is easiest.
• Times 7: These must be memorized. (Please add a comment below if you know a trick for these!)
• Times 8: Double the number once, double it again, and double it a third time. The result will always be even.
• Times 9: Two tricks here for multiplying single digits by 9. 1) The fingers trick: see http://www.multiplication.com/lesson10_nines_fingers.htm.
• 2) Take the number you are multiplying by 9, for example 7. The tens digit will be one less than the multiplier (6, in this example). The ones digit will be whatever it takes for the two digits to add up to 9. In this example, 6 + 3 = 9, so the answer is 63.
• Times 10: The concept here is that when multiplying by 10, it increases by one place value. So 1 x 10 is 10, 10 x 10 is 100, etc. Thus you append a 0 to the number, increases the value by one place. The result will always be even.
• Times 11: For 1-9, the result is always that both digits will be the same as the multiplier, for example, 3 x 11 = 33. For two-digit numbers 10-19, there is a cool trick. The first digit will be the same as the first digit of the multiplier; the second digit will be the two multiplier digits added together; and the last digit will be the second digit of the multiplier. For example: 11 x 18 = 198, 11 x 13 = 143, etc. Just sandwich in the sum of the multiplier’s digits between the multiplier’s digits to get your product!
• Times 12: Same as the 6 trick, but double the result.

Tips for Dividing or Factoring Whole Numbers

If you are trying to check to see if a number is divisible by another number, or can be factored by that number, or has a common factor with another number, these tips can be helpful.

• Divisible by 2: Any even number.
• Divisible by 3: Add up the digits as if they are all ones. If the sum is divisible by 3, the number is divisible by 3.
• Examples: 143 -> 1+4+3 -> 8, so not factorable by 3. 144 -> 1+4+4 -> 9, so factorable by 9
• Divisible by 4: If an even number is still even when you cut it in half, it is divisible by 4.
• Divisible by 5: Any number ending in 5 or 0.
• Divisible by 6: Any even number that fits the 3 rule.
• Divisible by 7: Memorize these.
• Divisible by 8: If an even number is still even when you cut it in half, and in half again, it is divisible by 8.
• Divisible by 9: Same as 3 rule, except the sum of the digits must be a multiple of 9.
• Divisible by 10: Any number ending in 0.
• Divisible by 11: See the Times 11 rule and reverse it.
• Divisible by 12: Any number that fits both the 3 rule and the 4 rule.

I also recommend Greg Tang’s book The Best Of Times, which is full of multiplication tricks like these, but in a fun, picture book format that children will enjoy experiencing.

A great video showing how to fill in a times table chart, and learn the facts while you are doing so, can be found here:
http://mathplayground.com/howto_learnmultfacts.html

If you have learners who struggle to learn their multiplication facts because they have trouble memorizing or can’t learn through these tips, or are just kinesthetic learners, try playing games where children toss a ball back and forth while skip counting with different tables. If they don’t know the tables at all, they can use a chart on the wall for reference while they are learning. Like with any skill, practice and repetition will eventually lead to mastery.

Tags: Multiplication tables, math, math education, Math facts

On Friday, my young writers and I had a Thank Goodness It’s Over party to celebrate our accomplishments during the month. The TGIO party is a well-established tradition for any NaNoWriMo group. I have always used it to showcase and celebrate each individual child’s writing.

We met at a family’s home, and each child had five minutes to read an excerpt from his or her story. I was impressed by the quality of the writing; three years of doing Nano for most of them has led to exceptional storytelling abilities in these young writers. During the reading part of the get-together, we had the usual stage fright issues, eventually overcome, and we had to practice being a good audience, also as usual.

The new experience was that one of the students had written a song and dance into his story, and he read us the description. His mother encouraged him to perform it for us, which he did. What followed was an expression of pure delight and joy as the boy rapped out a song and did a rhythmic, but hilarious, dance with it. We all laughed so much, along with him, that nobody managed to videotape the performance, but I did manage to snap a photo or two. That was an experience I will never forget.

After we all had a chance to read from our stories, we all enjoyed snacks, and the children played together. It was a perfect ending to a great program. The photo below is me with most of the participants (some couldn’t make it) holding their winner certificates.

Today is Wednesday, December 1, and November is finally over. All of the students in my program, Your Greatest Writing Adventure Ever, achieved their goals of writing a story in the month. The word count goals ranged from 1,500 to 4,700 words, and their ages ranged from seven to ten. What an amazing accomplishment! Not only the writing, but the fun they had doing it.

I am so proud of everyone, but I’m also relieved that NaNaWriMo is over. In fact, doing this program is such a mammoth accomplishment that it’s pretty much a requirement to have a TGIO (Thank Goodness It’s Over) party at the end of the month. I will be attending two: the party for my students on Friday night, and the New York City party for grown-up Nano-ers like me who wrote the full 50,000 words. Time to celebrate our accomplishments!

All the students in my program will have the opportunity to self-publish their illustrated works through StudentPublishing.com.

On that note, my own novel, a futuristic biotech thriller, doesn’t really inspire me to revise and publish it. However, I do think it would make a good screenplay, so my rewrite may be during Script Frenzy in April. That may also be my next group writing program, so please get in touch if you would like your child to participate!

Now that November is over, I should also be able to write for the blog more regularly again. I just did not have the time during this month. Writing 1,667 words per day in a story is quite a commitment! I’m glad for the experience, though, as it helps me relate even more to the struggles of my own student writers.

Imagine my delight today to find out from one of my young writers that my students’ anthology from last year had been featured on the NaNoWriMo Young Writers’ Program blog in late October. I’m kind of surprised we didn’t hear about it sooner, but it’s inspiring to discover it now, since so many of us are struggling with word counts and the challenge of finishing our stories by the end of the month.

You can watch the video below, or view the video on the NaNoWriMo YWP site. The anthology is called The Sun Shines on the Golden Dragon and the Mysterious Wizard, But Not on the Fat Smelly Alien. It is available for purchase on Amazon.com, and any royalties go to support The Garden Road School, a wonderful, progressive, non-profit school.

If you would like more information on the anthology, please visit the About Susan page of my website.

Tags: NaNoWriMo, ywp, Anthology, Creative writing

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!

I was at the Westchester County Airport this morning, in the women's room, when a woman and I started a conversation over soap. The topic soon changed to bathroom decor, which we discussed for a few minutes before wishing each other well.

Then the woman walked out, and I was able to see her gait. It was quite lopsided, and she walked with the aid of a cane. I had noticed that she had mostly been using one side of her face to talk, but now I could see the extent of the asymmetry of her body.

After seeing this, I found myself wondering why. A stroke, some kind of palsy, something else? Then I stopped myself. Why was I focusing on this? The woman clearly wasn't; she didn't even use the handicapped stall, though she would have been most entitled to use this spacious stall.

This thought process led me to think about education in general, and how much focus there can be on what is "wrong" with a student, to the point that all we can see are the "problems." Many educators have written about the issues around labeling, and I think this is what is at the core of the matter: that instead of a whole human being who may be encountering some challenges, children end up being viewed - and often viewing themselves - through the lens of a diagnostic label.

But those closest to them know they are so much more than that label. The families that have cared for the children since infancy and have seen their serious, funny and talented sides need to know that the teachers of their precious children will be seen as the whole people they are.

It's hard for teachers to do this when under the tremendous pressures of time, curriculum, large class sizes, behavior issues, and most of all, performance on standardized tests. It is also normal for our brains to observe, analyze, and try to understand things we encounter that are outside our typical experience.

If we want children to succeed, though, we need to override this urge to focus on what is “abnormal” and bring humanity back into the classroom as the main driving force behind what we do.

Tags: labeling

Yesterday was the first day of NaNoWriMo, and it began with a bang. My group has seven bright, eager children in it, and we all dove in to our writing projects yesterday.

Prior to that, we had a couple of meetings in which we worked on character development, understanding what plot is, setting expectations, and deciding on word count goals. I think my students from previous years underestimated their abilities yet again, if yesterday was any measure; they seem to grow their ability to write fluently almost exponentially each year. I’m impressed.

Even more impressive, one of our new members, a second grader, outstripped everyone in word count during a word war or two. This was the same little girl who couldn’t even get started at first. She was so excited and proud by the time her mother came to pick her up.

Like every year, I write alongside the students, and we all share excerpts from our writing in progress. Last year, however, I was writing a children’s book, while this year I’m writing an adult thriller. This means my word count will need to be higher, and I won’t be able to share all of it with the children. I’m also less enthusiastic about the subject matter; it was a plot idea that came to me months ago, and it’s just not as alive in me now. I started without any idea of characters, settings, or even specific plot ideas, so it was really stretching to get anything down.

On the bright side, though, I did reach over 1,700 words last night, the minimum to accomplish 50,000 words in a month, and the story wheels started spinning in the shower this morning. So maybe it will take on a life of its own yet again.

The write-ins are such motivators to get the ball rolling that I’m glad we held a meeting this Monday. We will meet again on Friday for those who want to get together. Be in touch if you’d like to join - it’s not too late!

Tags: writing, Children, literacy

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

I’m on my way to participate in the NCTM Regional conference in Baltimore, Maryland. My presentation is first thing tomorrow morning, and it will be on the topic of Singapore Math model drawing. The session is 75 minutes long, enough for a taste of several types of model drawing. Hopefully the participants will come away with some understanding of the power of model drawing and will be able to put it to use.

If you are there, come by and see me after the presentation, from 11:00 AM on, at booth 614, with SDE!

Tags: NCTM, Model drawing

A new reality show called Teach premiered on A&E on October 1. It follows Tony Danza as he enters the teaching profession as a high school literature teacher, with no prior teacher training.

I was fully prepared to dislike this program, as the preview indicated it would be another feel-good show about a former actor getting a chance to make a difference in young people’s lives. Why, I thought, aren’t they featuring an excellent non-celebrity teacher? But I was pleasantly surprised.

Teaching is a hard profession. It’s even harder if you want to be good or excellent at it. The show does a good job of portraying the real struggles that real teachers face: students who don’t understand what you are teaching, guardians who are in your face about their children’s needs, the demands of a curriculum and needing to differentiate your instruction to meet very diverse learning styles, administrators who bring you into their office for “the talk” about problems in your teaching, and more. It’s enough to bring even the strongest to tears, and I was glad to see the show doesn’t gloss over the pain and struggle of that first year of feeling so inadequate.

Tony’s tears and statements that “I’m not sure I can do this” are feelings I’m sure every teacher can relate to. In once scene, a special needs teacher told Mr. Danza that 100 teachers in their district had already dropped out of teaching in the first week, and she was glad he wasn’t one of them.

Danza clearly has a good heart and wants to do well by the students. However, experienced teachers will cringe at many of the novice mistakes he makes, ones that could have been avoided had he attended a good teacher training program and had an experienced mentor. For example, when he wants to talk to a girl about her poor performance, he does so in class, in front of her peers (and the camera), and he does so too long. Talk about embarrassing - no wonder the girl shuts down and escapes as soon as she can! Tony, you have to keep it real but light, and never in front of the friends. Avoid embarrassment and shame above all.

Another major misstep is how he handles special needs requests. The coach and administration try to push him into allowing students to go to the resource room on request by explaining the legality question. If the students request to work in the resource room, they have to be allowed to go. Danza doesn’t understand and instead keeps trying to build their self esteem, viewing them as lazy and able to achieve more if they only try.

The legality argument does nothing to convince Danza. The only thing that works is a special meeting to explain special needs. Towards the end, Tony is handed a piece of paper to complete. The page has text with mixed-up and backwards letters, as well as some non-words. When he is told that this is how some students see the text he is giving them, the penny finally drops, and he sees how wrong he has been.

If you would like to see this for yourself, there is a good introduction from the University of Essex that includes several reading comprehension tests simulating different types of dyslexia. I remember my special needs classes and how simulating life with a learning difference enlightened me before I ever entered a teaching situation. Without that kind of experience or training, teachers can do more harm than good, as the show demonstrates.

So is it a good idea to feature a reality show about a celebrity-turned-teacher? In this case, I think it is. Danza is personable, comfortable with the camera, able to be honest in front of others and on camera, and has the name to draw watchers. I can only hope that seeing some of his difficulties may show some of the non-educators who create education policy what teaching is actually like. Of course, they would probably think they would do a much better job. Who doesn’t, until they are tested?

Tags: Teachers, Tony Danza, Television, Special needs, dyslexia

Last Friday, this New York times article about Singapore Math appeared. The premise of the beginning of the article is that by studying one number at a time slowly, students learn more thoroughly and therefore build a better mathematical foundation. This is true, even if it is an oversimplification of the curriculum.

Here is a quote from the article:

Principals and teachers say that slowing down the learning process gives students a solid math foundation upon which to build increasingly complex skills, and makes it less likely that they will forget and have to be retaught the same thing in later years.And with Singapore math, the pace can accelerate by fourth and fifth grades, putting children as much as a year ahead of students in other math programs as they grasp complex problems more quickly.

This is true, from what I have seen and heard from different teachers. Not only that, but the mental flexibility for problem solving can be much greater with Singapore Math, if it is taught correctly.

And here is one of the main reasons I recommend this program:

Singapore math’s added appeal is that it has largely skirted the math wars of recent decades over whether to teach traditional math or reform math. Indeed, Singapore math has often been described by educators and parents as a more balanced approach between the two, melding old-fashioned algorithms with visual representations and critical thinking.

So you don’t have to sacrifice any of the important aspects of teaching math if you adopt this method. It does require some training and/or learning in order to implement it well, though, because the curriculum books on their own don’t offer a thorough grounding in the theory and practice.

What are your thoughts about the article?

An article appeared in the Lowell Sun yesterday, and this article triggered more questions than answers in my mind.

Among other things, the North Middlesex Regional School District found spotty improvements in math test scores. The article said:

Brady and Muir discussed how the district's use of so-called Singapore math is problematic. "We think Singapore math has taken us as far as it can," Brady said.

Muir added that Singapore math does not align with MCAS frameworks but that the district is looking at other textbook publishers.

This raised some red flags for me. Singapore Math took them as far as it can? I don’t think so, because the curriculum can take students very far indeed - if it’s implemented correctly. I’ve had third grade students tackling sixth grade problems with ease and confidence after using the program. I’ve also seen how far above the level of the math students in even high-performing school districts my Singapore Math students have been.

Not only that, but one of my earlier posts links to a longitudinal study in Massachusetts, the same state as this article covers, showing that Singapore Math does indeed raise test scores - the same test as the students in the Lowell Sun article took.

So my questions are: how much, if any, help did the teachers receive in implementing Singapore Math? What levels of textbook did they use, and were they the right levels for their student population? Was it a rolling adoption or done all at once, so that the students at the highest grade levels were left with the least foundation? What struggles did the teachers have, and what types of support were they given?

Appropriate professional development is necessary to implement any new curriculum well. If this district wants to switch to yet another curriculum, will they provide the training required to equip the teachers for student success? If not, they will just be setting the stage for another failure.

Tags: Lowell, Massachusetts, math education

For some people and classrooms, it just isn’t feasible to purchase high-quality number bond flash cards like you can obtain through Crystal Springs Books. I came across a site where you can download your own, print them and cut them out. These are addition and subtraction cards appropriate for grades K-3. You can download them for free after registering at Teachers Pay Teachers.

A caveat: While the free cards by William Hughes cover the basic principles of number bonding for addition and subtraction for numbers to 10, the Crystal Springs cards are better because they do not rely on just one position of the whole vs. part numbers. In other words, on the Hughes cards, the “whole,” or total number, is always on the left, while the two smaller numbers, or “parts”, are always on the right. On the Crystal Springs set, the whole can either be the top number, the left number or the side number.

I think this is an important teaching tool for students to understand that an equation is not directional, but must have equal parts no matter which direction the equation goes. When students are taught that equations always go from left to right (which actually happens in some elementary classrooms), for instance 2 + 2 = 4, then they often flounder when they encounter pre-algebra and are expected to solve problems like 6y + 6 = y + 26.

So the earlier we teach that equations are like a balance scale and not like a one-way street, the better our students will be equipped when they encounter concepts like these in algebra.

Tags: Singapore, Math, singapore math, math strategies, numeracy, success, number sense

I’ve been reading blog posts about math like this one, in which a common theme is that we need to return to “basic” or “regular” math skills and dispense with the constructivist programs that were so popular in the last decade or two. I’ve read about people who are frustrated with the idea that children should have to reconstruct the math theorems that evolved over the last 2000 years or so, and it makes sense that they shouldn’t have to do this. Some people turn to Saxon Math, which I have taught, or Singapore Math, as the solutions that teach the basics and provide a strong foundation for children to learn math.

All of that is fine, but what I challenge is the idea that you are not involving constructivism or critical thinking when you use Singapore Math. If you are teaching Singapore Math well, or any math program for that matter, it should involve a great deal of critical thinking and metacognition, or thinking about thinking. The children should be asked, and learn to question themselves, questions like, “Why did you choose to solve a problem this way?” or, “Why did you choose this mental math strategy over another?” and, “Why does this math algorithm work? Explain it to me, or explain it to your friend...” and so on. All of which strengthen the math constructs inside a child’s mind and demonstrate conceptual understanding inside themselves and to others.

So to my mind, constructivism and Singapore Math are not at all mutually exclusive. It’s the approach, and the child’s resulting numeracy, that matter in the end.

What do you think?

Tags: Constructivism, Singapore Math, math education, TERC, Everyday Math, Connected Math, reform math, fuzzy math, Numeracy

A New York Times Magazine article titled Building a Better Teacher appeared last March. It’s an excellent contribution to the debate about what makes a good teacher. As the article describes, it’s not enough to care a lot; there are many caring teachers who can’t get their students’ attention to teach them anything. Being a good teacher is not strongly correlated with the graduate schools they attend, their teacher test scores, or particular personality characteristics. None of these predicts teacher effectiveness well. Merit pay and high pay incentives, haven’t worked to improve teacher quality (or test scores) either.

In fact, it is so difficult to quantify what makes a good teacher that the latest strategy is to throw out all the “bad” ones and keep all the “good” ones. That comes from the notion that there is some sort of magic that can’t be taught that makes a good teacher.

But the article describes two major issues regarding teacher education. One is the lack of explicit classroom management instruction, something every teacher I have met bemoans about their teacher training program. Doug Lemov, who helped found Uncommon Schools, traveled around the country to observe and record the techniques master teachers would use to manage their classrooms. Rather than some kind of “magic” or innate genius, these were techniques the teachers were often conscious of implementing, but they were so good at them that they looked like magic. He compiled them into a taxonomy and implemented them in the Uncommon Schools training program. After going through this training, apparently even first year teachers demonstrated classroom management mastery.

I can testify to the importance of explicit teaching in this area. Unlike many teachers, it seems, I had the benefit of excellent mentorship by Charles Fischer, among others, at a private school at which I made special arrangements to do my student teaching. It was such a valuable experience, and I took away many new arts and skills to help survive even very difficult teaching environments.

The second major issue is to do with content area teaching. The article focuses on math, which doesn’t surprise me. A teacher can have excellent management skills and not have a strong grasp of the content areas he or she is teaching, and teacher tests aren’t enough to assess that. Deborah Loewenberg Ball started a research project to look at the specialized skills in teaching even elementary math, which include not only understanding how math works, but why certain misconceptions would lead to children’s mistakes.

She also developed a test for Mathematic Knowledge for Teachers, or M.K.T. Scores on this test did translate into predictions of effective teaching. However, the question of how to teach teachers so they do score well on this test remains. In my experience, this kind of critical thinking is an essential part of teaching and learning Singapore Math, and strong training in teaching this program can really help a teacher with the understanding they need to be an effective math teacher.

What are your thoughts?

Tags: Teachers, Teaching, teacher quality, merit pay, classroom management

The Urban Word of the Day today was “cerebral bulimia,” defined as “binging and purging of the brain.”

Doesn’t that sound familiar? It calls to mind all the useless studying in which facts are crammed into temporary storage in the brain, dumped out on paper (or computer) for a test, and promptly forgotten. This can, of course, include math algorithms and formulas.

How different it is when, like an athlete learning what works best for her body, we learn the fundamental concepts behind new ideas first. Then the algorithms and formulas follow logically. It can even be possible to recreate them if they are forgotten. This builds long-lasting skill and “muscle” to handle even the most difficult challenges.

Tags: learning, study skills, conceptual understanding

New York State is putting teachers and students under more pressure by revising the required scores on standardized tests for students to achieve proficiency.

Why are they doing this? Apparently it is because despite passing Regents tests, almost 25% of the students need extra support once they reach college.

How will they afford “remediation” for all the passing students who are now failing?

One more sign of why it is so important to put conceptual understanding ahead of rote application of algorithms. It is easy to forget what one learns by rote, but not deeper conceptual understandings.

Here is a link to the abstract of the original article. Regretfully, I did not save the full copy when it was current.