Archive for September, 2010

Free Number Bond Cards and Thoughts About Equations

Wednesday, September 29th, 2010


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.

Pluses and Minuses of Singapore Math

Wednesday, September 29th, 2010


This homeschooling website has an article about Singapore Math that is short but informative. It tells a brief history of Singapore Math in the US, then goes into why it may or may not be the best choice for a homeschooling curriculum. Here is an extract from the article:

The curriculum uses a true spiral approach, a method that is used less successfully in the United States. In the spiral approach, the curriculum assumes prior mastery of the subject in the previous grade and so does not review basic processes but moves on to a higher level in each subject. If single digit addition is learned in grade one, then it will not be reviewed in grade two. Instead, double digit addition will be introduced and taught.In the United States, the student generally has a review of basic concepts each year before moving on to the next level of learning.

I think the article’s points are good, with the exception of the idea that all of the curriculum is consumable. With the textbooks, it is possible to reuse them (though as paperbacks, they can take a beating) and have the students copy their answers into math journals or separate pieces of paper.

Constructivism vs. Singapore Math

Tuesday, September 28th, 2010


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?

Report: Singapore Math Works

Monday, September 27th, 2010


There is an interesting website devoted to reforming math education in Utah; it is On it, among other text and video content, there are links to reports about a longitudinal study in a Massachusetts school district showing the improvement in test scores over the long term when the district adopted Singapore Math. The evidence is compelling, not only qualitatively but quantitatively. I have also provided the report for download here.

Video: Singapore Math Training for Parents

Monday, September 27th, 2010


If you are a parent who is interested in how Singapore Math works, but you don’t have time to attend a training session or do a lot of reading about it, here is a resource for you. This school recorded a session in which their parents were taught how this program works and how to support their children. Part 1 of the videos is here; click the video to go to Youtube to view the other parts.

Measuring Teacher Quality: Classroom Management vs. Content

Monday, September 27th, 2010


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?

Yet Another Private School Adopts Singapore Math

Sunday, September 26th, 2010


It’s a measure of what a difficult situation our US schools are in when the math program a school adopts makes the news. But that is exactly what is happening often now that our nation is recognizing how far behind we have fallen in math and science.

This article outlines the reasons this Chester, PA private school is adopting Singapore Math for their lower school. I have included it here in PDF form so I don’t lose the link when it is archived. The reasons are pretty solid reasons as to why any school should at least consider this program when looking to improve their math education.

A New Story Begins: This Year's NaNoWriMo Program

Saturday, September 25th, 2010


This is the first year since 2003 that I will not be teaching NaNoWriMo in the classroom. What’s good about this is it frees me up to do it with a whole bunch of area students. I have posted a new program description for our fun, exciting adventure to come.

Here are a few details not included in the program description page:

Each student will get to set his or her own word count goal, usually with some consultation with me. That goal can be fairly flexible to a point in November. The idea of the program is to write as many words in a story form as possible, creating fluency and breaking self-imposed limitations of what we think we can do. The important point is to meet the goal. Editing and improvements can happen later.

As a several-time NaNoWriMo participant and winner, I am in a good position to empathize with the struggles and triumphs of this challenge. I also write alongside the students, so rather than a taskmaster, I am a fellow writer and friend. We inspire each other.

Last year, with my amazing class, they did so well and were so prolific that I had to find new ways to encourage them to continue as they surpassed, then doubled, then tripled their original goals. Everyone surprised themselves, and me. I finally challenged them to a race, and the first person (me) or group (them) would win a prize from the other. It was neck and neck until the very last day, when the group emerged victorious. I treated them all to hot cocoa at the local coffee shop a few days later.

An added dimension this year is that I am working with the YWP to develop teacher training, so even more students can experience the joy, delight and struggles of this roller coaster ride. I welcome contact from teachers who are interested in finding out what it can do for your students.

Sign up for the program now, and find out more about NaNoWriMo’s YWP here.

Dyscalculia and Teaching Math

Thursday, September 23rd, 2010


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.

13thwinterIf 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. 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.

"Cerebral Bulimia"

Wednesday, September 22nd, 2010


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.


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