Bar modeling, the technique of solving math problems using a strip or bar, has become synonymous with Singapore Math. The math program in Singapore uses such technique to solve a large variety of math problems.
But the technique has been around for many years as evidenced by this arithmetic textbook that I recently found, published in 1919. (Hamilton’s Essentials of Arithmetic)
traditional math 
What’s also not new under the sun is how different variations and wordings confuse students. Discovery learning is also not new, but I’m all for the version that requires individual “micro” discovery, homework problem – by – homework problem, night after night using proper textbooks. Most of my discoveries came that way and the rest came when I was directly taught by a teacher who knew the material and didn’t waste our time.
In-class student-led group discovery is just soooo tedious and the best students HATE group work, especially group grading. The poor students go along for the ride and there is absolutely no evidence that curiosity or engagement has any effect on improving, let along creating mastery and the ability to actually DO other problem variations themselves. Project-Based Learning teaches vocational hacking skills, is bad for groups where students can hide and do little work, and is especially bad when students in the group assume roles, as with STEAM. My coaching experience for my son’s FIRST Lego League gave me proof of that.
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Unfortunately along with these successful methods being around for over 100 years, so has progressive ed techniques. Books written from same time period, follow John Dewey’s child centred approach in BC’s math curriculum.
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Yes, but even in books written by the progressivists and Dewey-followers of those times, there were plenty of practice problems, and there was usually a summary of the finding that was to be discovered. Also, the problems were structured to allow for such discoveries to be made by way of the “worked example” effect and carefully scaffolded problems. As I mentioned in my talk at the ResearchED conference last summer at Oxford, my arithmetic books were co-authored by Brownell, considered a big math reformer of his day, and whose name is still touted as a pioneer in this even by NCTM. Nevertheless, the books have many of the earmarks of traditional instructional techniques that have been decried as ineffective and “having failed thousands of students”. There were many practice problems along with solid explanations of the underlying concepts. Thus, “carrying” and “borrowing” were explained so it wasn’t merely a rote learning exercise with the instruction of “just do this and don’t ask questions” as is intimated by reformers.
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What age/grade level was this text intended for?
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Looks like 7th grade level.
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I remember the Everyday Math approach my son had called “Remember the One.” It was inane. But worse was their limited coverage before moving on. They assume that kids have the attention spans of chipmunks. They only expect kids to do 4 or 5 homework problems before moving on and I remember that the teacher didn’t much care about mastery because EM is all about spiraling (cough … circling). However, students never learn the material enough to build a scaffold. EM is not spiraling, it’s repeated partial scaffolding. I had to ensure mastery and scaffolding of the material at home so that EM became a spiral and not a circle. You might see some decent math examples in EM, but the process is all screwed up and very dependent on parental support. Schools now expect parents to be teacher helpers, go to their open houses to learn their math, and “practice math facts” at home. That just increases the academic gap.
There are many subtle variations of how you encounter percentage problems and it takes a lot of consistent individual homework problems to make sure that you build the proper level of understanding. EM expects you to build this understanding over years. One parent I talked to was SO angry because 3 of her kids in three different grades were covering almost the same level of material. This appeals to schools that use full inclusion because they think kids will magically learn when they are ready – as if “deep conceptual understanding” is some magical sort of IQ epiphany process. No. In fifth grade, my son’s teacher had to NOT trust the spiral and enforce mastery of the basics because a number of bright kids had not yet mastered the times table – and their parents finally came to their senses and were sooo pissed off.
What if a suit costs $350 and is “discounted” by 25%. Then you get an “additional” 25% off. What is the final cost of the suit? Understanding is complicated by vagueness and the difficulty of converting words to equations. That skill is developed by doing a lot of homework problem variations. Tests, like the SAT, AP calculus, and especially the AMC test go out of their way to find problem variations students have never seen before. Solving them takes a lot of work on individual problems because getting a good grade is all about making small sideways transitions or “discoveries” rather than major leaps of discovery down from some vague and general conceptual knowledge. That doesn’t happen. That superficial belief just allows modern educational pedagogues to think that they can get full inclusion for free with lower expectations and more understanding. Only in their fuzzy dreams. For most unsupported students, it’s all over by 7th grade.
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Steve. Just curious about your feelings regarding the old Saxon texts. Saxon’s underlying belief was that it takes several weeks of practice before a child will master a concept. He delivered a string, explicit lesson initially, offered 5 or so homework problems on that concept that day, and then provided 3 or so practice problems in each assignment for weeks afterwards. He cringed when anyone referred to this as spiraling. He called it continual review.
This made sense to me, mainly because it is how I learned best. My students did well with it.
I’m interested to hear your perspective. I’m sure you know about it.
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I can’t speak for Steve here, though I suspect he will agree with me regarding Saxon. Call it “interleaving” or “distributed practice” or whatever, it is an effective method for learning. It ensures that students are continually practicing procedures/problems that may not be the subject of the particular unit being taught. Also, there is a tendency in some books to give students a block of the same type of problems, like “What is the amount of discount on a $200 coat reduced by 25%”. Mixing different types of percent problems so they have to grapple with the different variations, what is being asked, and how to solve it, are focused on.
I think Saxon (at least the old versions) is effective. I chose to use Singapore’s math program with my daughter, however, because I needed to make up for lost time (we started in 6th grade unfortunately) and really hone in on fractions. Saxon’s continual practice format doesn’t allow for such concentrated surgical strikes, and you are held captive to the program. Once you start you have to finish, and I didn’t have the luxury for that kind of investment of time.
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