Review: Discovering Mathematics (Singapore Math, Secondary Level)

Note: Since beginning Discovering Mathematics, Singapore Math has released a new edition, Discovering Mathematics Common Core. The order of lessons vary a bit, and new topics have been included. At this writing, only levels 7A and 7B are available, with 8A and 8B to be released at the end of 2012. The others should follow soon, I’ve been told. Yes, movement between the old and new programs are doable, and a switch from  the old level 1 to the new level 8 works well, I’m told by their forum guru. Worrier that I am, I have 7B and 1A just to be certain we won’t miss anything. In all regards, including, rigor, the new books are quite similar to the old, with additions and a bit of rearranging being the main differences. 

Providing a challenging mathematics education was one of the key reasons we started homeschooling. Deeply disappointed by the depth of the math provided by two schools, my older son, then seven, assumed he was the problem.

“I don’t think I’ve very smart, Mom,” he told me.

“Why not?” I inquired.

“Because they don’t give me anything hard to do,” came his sad reply.

Math (and science) were his loves at age 4 and 5 in Montessori and while at home. He was appropriately challenged in the first at school and free to explore the second at home. First grade ended all that, where math became repetition of previously mastered lessons. Second grade, at our local gifted and talented public school, it was nonexistent  which was because, we were informed, he knew all the material for that year already.

So once home, math took a starring role. Singapore Math quickly became our preferred curriculum (reviewed here) for the elementary sequence. Even doing the Challenging Word Problem books, we burned through it quickly. Almost 10, my older insisted on Algebra, so we started the standard sequence, happily making our way through a fine text, Jacobs’ Algebra. (reviewed here).

When my younger finished 6B, I wondered if there was another way. We vamped for much of last year, working through a variety of books while choosing our next course of action. After much consideration, we decided to stay with Singapore, specifically, their Discovering Mathematics series. This four-year series is designed to cover some prealgebra, algebra (I and II), geometry, and a smattering of other topics, like probability and counting. Unlike most American programs, these topics are interwoven throughout the years, with chapters on algebra followed by chapters on geometry with a side trip to data handling. It’s challenging, with plenty of problems, tests with answers, and teacher’s support books if needed.

But I hesitated. Accustomed to the four-year math sequence I’d known as a child and that my older son had followed, I was hesitant to commit to a different path. What if we didn’t like it after a year? What then? (Answer: Start a traditional Algebra program and compact or test out of what has already been covered. Ditto the next year with Geometry.) I presented my younger son, then 10, with the options. Singapore, Jacobs, or Art of Problem Solving? He looked at samples of all online and liked the familiarity of the Singapore. Thus, we reached a decision.

We’ve not been disappointed. We started Discovering Mathematics 1A soon after it arrived and found that while it certainly felt like the Singapore Math we’d enjoyed the previous years, it was a step up in challenge and pace. He’s enjoying it, but we don’t whip through the pages as we did at the elementary level. Concepts aren’t broken down in such small parts, and even the sample problems (Try This!) are fairly challenging. Fortunately, this increase in challenge has resulted in an increase of effort. As a result, he’s feeling rather accomplished while learning large amounts.

At the minimum, the user will need to purchase two textbooks for the year. These paperbacks are affordable and reusable, in keeping with Singapore Math’s reputation for affordability.   Each of the four levels requires two textbooks, each generally over 200 pages long. The year is broken up into 11 to 17 chapters, roughly evenly divided between the two books. (The fourth level is shorter, with a significant proportion of 4B dedicated to review tests, similar to the elementary level 6B.)

The chapters are broken up into shorter sections, some amenable to a single lesson or day of work, others requiring multiple days, given the depth of the lessons. Each section ends with problems in four categories: Basic Practice (the easiest problems), Further Practice (definitely a bit more work), Maths@Work (word problems just as challenging as the aptly named Challenging Word Problems of the elementary series), and Brainworks (sometimes too hard for Mom but worth trying if no one is crying). The so-called Revision Exercise (test) at the end of each chapter is at the level of the Further Practice and Maths@Work level. Aside from the Brainworks problems, all the answers for the problems are in the back of the book. If you desire worked solutions (and so far, I’m good without), there are Teacher’s Guides available, which include other teaching assistance, activities, and a breakdown of lessons and timing.

An additional workbook is available for each level, providing some extra practice as well as more problems at the more challenging level. Unlike the traditional workbook, these don’t provide a place to do the problems, making them more of a reusable problem bank. I assign some of these at the end of each chapter, before the revision (test). The number I assign depends on how well he’s handling the material — some sections just require more practice than others. Generally, these workbook problems are more challenging than the textbook ones. They are broken down into sections called Basic Practice, Further Practice (both a bit more involved than the same-named section in the text, it seems), Challenging Practice (and it generally lives up to its name), and Enrichment (excellent problems that we don’t get to most of the time). As with the text, answers are in the back, but solutions require the Teacher’s Edition of the workbook. I’d strongly suggest the workbook to supplement all learners, with the Teacher’s Edition on the shelf if a parent is a bit math wary and wants guidance on the trickier problems.

The strengths of the elementary level of Singapore Math continue at the secondary level. The pace is swift, which is excellent for the mathematically talented child but could be overwhelming for others. The problems in the text at the secondary level are far more challenging that what is in the workbooks for the elementary level, but on par with the Challenging Word Problems books. (I’ve not used the Intensive Practice books at the elementary level, which are designed to increase the challenge at their respective levels.) The depth we’ve encountered thus far is also impressive. Math is not taught via algorithm but by deep understanding, which, in my opinion, is by far the superior method. It is applied, not simply in one-step word problems, but across the sciences and into the work world. Math lives in these books, with all its complexity and beauty there for the learning.

The downside to the Discovering Mathematics series? If one isn’t math-comfortable, these could be a challenge to teach. That said, for the math-uncomfortable, these are an excellent way to build a new relationship with math. I know that throughout teaching even the elementary level of Singapore Math to my boys, this math-comfortable mom moved from number capable to number savvy. I’ve said before that I believe that math is best taught rather than learned solo. Discussion is part of the process, and many times, I’ve had a child teach me and correct me, thus delighting the child and enlightening me. (For more on thoughts about strong mathematics programs, read my post, Math Matters.)

We’re early in our exploration of this four-level series, and I’ll post again as we move through the program. I’m hoping we continue to enjoy Discovering Mathematics over the next several years, allowing us continuity with a strong mathematics educational program.

As always, I only review what we’ve used, and I never accept compensation of materials or money for my reviews. 

Math Matters: A Response to “Is Algebra Necessary?”

Math has taken another hit. This time, it’s from Andrew Hacker, emeritus professor of political science at Queens College, City of New York, whose piece “Is Algebra Necessary?” appeared in the July 28, 2012, New York Times Opinion section. His premise is that since so many kids and adults find math hard, it just shouldn’t be required. Or at least, they shouldn’t be subjected to the algebra, geometry, more algebra, calculus sequence that had been the US math sequence for so long.

Here’s Hacker’s initial thesis: “Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower.” His support rests on statistics of high school dropout rates (75% of ninth graders complete high school) and the role he sees algebra playing in that rate. I have no doubt algebra is a serious stumbling block for many students.  Does this mean we should toss it out and teach calculating the Consumer Price Index instead, as Hacker suggests? No, although some shift in how we educate our children in math is definitely in order.

First, let’s clarify our terms. Arithmetic is what you were taught before algebra, at least in most schools and certainly what you experienced in grade school if you’re now in your 40s. Addition, subtraction, multiplication, and division (including that long kind Hacker says is essential). Fractions, decimals, and percents appear, with a nod to generally single-step word problems for application. Arithmetic is number-crunching. It can be taught with algorithms and tricks or deep conceptual understanding, although it’s too often taught the former way rather than the latter.

Arithmetic is not mathematics. Math is problem solving and reasoning, relying less on rules and more on understanding the way numbers and geometry works. Math uses those skills in arithmetic to problem-solve and reason, but not all arithmetic needs to be in place before math is taught. In fact, the best arithmetic teaching puts math at its heart.

It’s been said that arithmetic is to math as spelling is to writing. You’d not mistake a great speller for a fine writer, and while a fine writer should produce a product with conventional spelling, one must not wait to spell all the words one could ever want to use in order to start writing. We teach our kids to write before they can spell well, allowing both to develop together.  The same relationship can and should exist between arithmetic and math.

Math matters. Like it or not, some degree of math comfort and skill is mandatory for full participation in today’s world. A sound mathematical education opens doors into the STEM jobs that increase in number by the day. A good understanding of numbers makes economics, politics, and social science more accessible.  It allows people to understand credit, mortgages, taxes, and savings.  If we drop our current math requirements in high school, shifting them to simply consumer math, as it was called in my high school years, we vastly under-prepare our children for the options ahead.

Hacker agrees that math matters.

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey. (Hacker)

But then he backpedals, wondering why we subject our “potential poets and philosophers” to the traditional math sequence rather than nurturing their talents. My answer is because we really don’t know at 14 what path many kids will choose. We need to prepare them for more options that whatever their fancy of the week may be. I agree with Hacker’s assessment that today’s math path is likely outdated. I’d prefer to see more integrated math, weaving algebra and geometry together and focusing on applications across the disciplines.

What Hacker fails to address is the why behind the algebra fear and failure at the high school a college level. Early elementary math matters, whether at home or at school. Math (or actually arithmetic) taught as rote memorization with no deeper applications is like teaching spelling without writing, scales without true music, or love via self-help manual rather than personal experience. It deadens math. Kids aren’t born fearing and hating math. Aversions to math are learned, often from those teaching them math. Hacker spreads this sense of math as too hard in his final paragraph: “This of math as a huge boulder we make everyone pull, without assessing what all this pain achieves.”

What if, from the start, we taught our children to look at math with wonder. What if we nixed the “drill and kill” and senseless algorithms in favor of teaching actual math. I picked Singapore Math for a reason (reviewed here). It teaches math along with the arithmetic. The whys of the processes are right there, and the word problems (especially those from the Challenging Word Problem series) demand complex problem solving. There’s no waiting until algebra to think in math fluently. It starts in the first books. Singapore isn’t alone in this process. The arithmetic my sons learned at age 4 and 5 in Montessori was rooted in deep understanding of how the numbers work. Heavily manipulative-based and backed my a masterful teacher, they learned WHY regrouping works for subtraction and what is truly going on in long multiplication problems. (Note: Manipulatives are not the key here. The key is excellent instruction about the concepts paired with judicious use of manipulatives, followed by transitioning to abstract work without those hands-on items.)

Curriculum matters. What matters more is teachers (parents of homeschoolers, that’s you or whomever to entrust your children to for their math education). The best curriculum in the world won’t teach mathematical thinking if the educator is uncomfortable with the subject matter. Hacker’s treatise focuses on high school mathematics education, but he’s focusing on the wrong end. Those math teachers went into teaching to teach math, a subject they likely deeply appreciate and understand. But what about our elementary kids, at home or at school? While I know there are exceptions, I sense the comfort level with math (not arithmetic by algorithm) isn’t high for many elementary teachers, and I know it’s not present for many parents. (That’s why Saxon sells so well. It’s easy to use, highly scripted, and able to be turned over entirely to the kids early. And it’s arithmetic. Not math.)

So what to do? If you’re a homeschooling parent, find a curriculum that supports deep mathematical thinking. If you kids are young, rejoice. You can easily learn along with them. Singapore Math and Mammoth Math are fine elementary programs. Art of Problem Solving offers excellent texts for the set ready for pre-algebra and beyond, and Singapore has an integrated math progression for the post-elementary level. Too intimidating? Consider adding in supplemental math into your existing arithmetic program (see list below). Anything that demonstrates the wonder of math over the tedium works toward supporting young learners of math.

Hacker makes a few good points. Teaching more practical mathematics in high school makes sense. Not everyone will choose a path that requires calculus, and that’s fine. But rerouting the mathematical train before algebra is begun seems foolhardy and shortsighted. At 14, the age many American children face algebra (and I’d argue that’s way too late for math to truly begin), we don’t allow kids to drive, vote, sign a contract, or work most jobs. I’d add to that list we shouldn’t allow them to limit their career options by opting out of algebra either. Instead we should be making math meaningful during the elementary years rather than simply teaching rote arithmetic skills while bemoaning how hard math is. As parents and teachers, we can emphasize the utility of math and encourage deeper mathematical thinking skills, both by curriculum choice and in supplementary reading and discussion. We should not abandon algebra nor the rest of higher math. Rather we should provide proper support from the start to make that math not only possible to do but delightful to explore.

More reading on mathematics education:

• U.S Teachers Not Well Prepared to Teach Mathematics, Study Says
• I’m Bad at Math, and I’m Fine with That!
• Adding it Up: Helping Children Understand Mathematics (available to read online)
• Arithmetic for Parents (Ron Aharoni)

Supplementary resources for all ages:

• Calculus By and For Young People (Don Cohen)
• Challenge Math (or any title by Edward Zaccaro)
• G is for Googol (David M. Schwartz)
• The Man Who Counted (Malba Tahan)
• A Gebra Named Al (Wendy Isdell)
• The Number Devil (Hans Magnus Enzensberger)
• The Adventures of Penrose the Mathematical Cat (or any other book by Theoni Pappas)
• Sir Cumference of the First Round table (or any title by Cindy Neuschwander)
• Anno’s Mysterious Counting Jar (or any title by Mitsumasa Anno)

Preliminary Planning for 2012/13: My Older (10th grade)

A few weeks back, I posted preliminary plans for fall for my younger son, A.B. My older son’s plans still have some holes, but here’s what I have so far.  As always, plans are subject to change. For past plans for both boys, see the tab above, “What We Say We’re Doing.”

A.D. (15, 10th grade)

Math: This one is easy, at least for me. My older will be enrolled in a local homeschool-friendly university for Calculus I and II this year. Math is his strongest subject, and his biggest challenges will be showing his work and writing legibly. The math part should be no problem, especially since he’s spent the past month working through my college calculus text with the help of Khan Academy videos. Yes, he’s excited, albeit in that somewhat cool, detached way teenage boys often have.

Language Arts:  The goals for this year are to continually build his writing skills, with a focus on the essay and academic writing, and doing more formal literary study and analysis. For the writing, we’ll selectively work through Models for Writers: Short Essays for Composition, adding in a few research papers throughout the year.  He’ll also complete a Hewitt Lightning Literature course, likely American Mid to Late 19th Century, although that’s under debate.  I’d like to add some formal vocabulary study, since that fell by the wayside midway through Word Within the Word II. What we’ll use remains unknown (suggestions welcome).

History: This one’s a mystery. This summer, he’s watching and discussing The World Was Never the Same: Events that Changed History (Teaching Company) with a group of homeschoolers.  I’m adding some readings to round out the subjects as well. A friend is musing about creating a course on the history of the English language, but this is still in the maybe stage. Last year was American history, so this year won’t be. Beyond that, I’m uncertain (and again, open to suggestions).

Science: Ack! It’s physics time! Somehow, I found myself volunteering to teach (algebra-based) physics to a handful of local homeschoolers. Then, I promptly lost a night of sleep in sheer panic. I’ve found my ground and some good resources. We’ll likely be using Singapore’s Physics Matters for the text, with additions for the material it’s lacking (parabolic motion, centripetal and centrifugal force, and quantum physics, just to name a few). A friend’s husband, who will be doing labs for my younger son and his own son, volunteered to run labs for the high school kids one day a month. I’ll teach the material, likely working some smaller labs and demos during our weekly meeting, and turn over the true excitement to him. Lesson plans will appear on this blog as they develop.

Foreign Language: Latin didn’t work. Spanish with Rosetta Stone (assigned to give the flavor of the language only) yielded less than 20 vocabulary words, per my son’s estimate. So this fall, we’re trying something different. So this September, my older will start the American Sign Language  sequence at our local homeschooling-friendly university. A kinesthetic language for a kinesthetic learner seems appropriate. Will colleges accept it? Many do, and he understands the limits this choice may place on his options later.

The Rest:  He’ll continue with piano through the summer and next year. His negotiations with him piano teacher did yield a happier student and a generally satisfied teacher, and he’s pleased enough to stay put.  While he spent some time at tennis lessons this winter, he’s not interested in our local Y’s current configuration of classes (teens were moved from  classes with adults to classes for age 8 and up). He needs exercise, and finding what will work for him is one of our summer quests.

I’d like to teach a class using David White’s The Examined Life: Advanced Philosophy for Kids, although not first semester when I’m settling into physics. Both my boys enjoyed White’s Philosophy For Kids: 40 Questions That Help You Wonder About Everything, but I’d rather run this second one with a larger group than my own two children (and I think only my older is ready for this much more challenging tome).  I’m also waiting to hear from my older. It is, after all, his education.

Suggestions are always appreciated, as are links to your plans.

Show Your Work

Show your work.

If I had a refrain, that might be it. Sure, there are others. Take care of your dishes. Check your list. Shower now if you can’t remember when you last washed. Take your feet off the wall. But over the course of seven-and-a-half years of homeschooling, this one line may be the one that I repeat the most often.

Of course I say it most often about math assignments. I sometimes wonder if I say it so much now because I didn’t say it at all to my older when he first came home in the middle of second grade. The gifted and talented school he attended didn’t offer him math lessons, since he’d already met their benchmarks for the state’s standardized testing. You read that correctly. The school dedicated to gifted and talented kids didn’t math accelerate. It’s part of the reason we came home. Math and science were his passions, and school offered little of either. But I digress.

Once at home, I came face-to-face with the severity of his dysgraphia. His writing disability made lining up numbers near impossible, and even with large-grid graph paper, his fingers tired long before he made it through even the shortest problems. So I scribed for him. Reducing the writing load for the dysgraphic child frees the child to think about the work rather than think about forming the letters and numbers. So for years, he sat next to me, telling me what to write for problem after problem.

Well, not always. Often, he could do them in his head. For many of the word problems, he juggled numbers and concepts while walking around the room, producing the correct answer while I scribbled my own process down on scrap paper. He was often faster than I, and he was usually correct. However, he could rarely tell me exactly what he was doing up there in his brain, but since it turned out reliable results most of the time, I didn’t press him. On the times I did ask, the answers as to process were so convoluted that I couldn’t have begun to put them onto paper.

I wish I’d tried. He started Algebra just before turning ten. The math was a breeze. The new requirement that he write down each step was not. I worked problem after problem on the white board, showing each step. On his first attempts to isolate the variable, he tried to do what I had done. While he could easily find the right answer, his steps were scrambled versions of mine, unrelated to his answer but rather inaccurate approximations of what I’d done in the examples. I was stumped.

Eventually, as the problems became longer, he saw some merit to showing his work. There just was too much to hold in his head once he was factoring polynomials and the like. Besides, he’d discovered the wonder of partial credit.  I started giving him tests, hoping to coax some respect for accuracy out of him.  That did improve, but an unanticipated side effect was that he showed more work. If he worked an equation correctly but made an arithmetic mistake along the way, he could still gather most of the points for the problem, process being top priority. Show your work equaled a higher score, even with an addition error here and there. It wasn’t my intention that these be linked, but the result was desirable — he started to show his work more consistently.

Fast forward to Chemistry. Teaching two boys entering their teens (one wasn’t mine), school work often seemed to be an impediment to their good time. I started the year singing my “show your work” sweetly and gently. By November, it was more gruffly growled than sang, and I started deducting points on tests for non conformers. In general, that improved how much work was shown, but they were hardly showing it all the time. I developed a litany of reasons to show work. Here’s today’s version:

• When you show your work, you can more easily check your own work for mistakes. (Yes, this implies that one would naturally want to assure and answer was correct. No, this has not yet gained a convert to work-showing.)
• Writing down each step of your work helps you work more efficiently. It’s hard work to hold a bunch of numbers and variables in your head.
• Writing down work means you may get some points for a wrong answer. (AKA, the Partial Credit Plea).
• Showing your work tells me that YOU did the work, not the answer guide at the back of the book. (Yes, playing the “you could be cheating card” is rough, but that’s actually one of the more compelling reasons to show it for my older.)
• If you intend to be a scientist or mathematician, you must show your work so others can try to replicate your findings and verify that you did not pull the data out of your nose (or other body part). (Since my children aren’t inclined in these directions, this receives either polite nods or blank stares.)
• I’ll send you back to do it again (and again) if you don’t show the work, or I’ll simply mark it entirely wrong on a test. (Tough love or just practical? I don’t care. This one works moderately well, at least until said student just wants to be done and forgets this extra-work producing rule).
• Show it because that’s what you’d have to do in school. (Yes, this one seems lame. However, my older returns to the classroom for Calculus in the fall, so the skill needs to be in place).

I’d love to report that I don’t need to sing or scream that refrain anymore. My older’s heard it so many times that I’m surprised he doesn’t mutter it in his sleep. Yet just this week, he was sent back not once but twice on the same assignment with my chorus ringing across the house. My younger ran into the same chorus an hour after the older’s first offense. He repeated the assignment, this time extolling the virtue of showing work and promising to sin no more.  I’m cautiously optimistic.  Until I know he really is a true work-shower, I’ll continue to remind him before problem sets and likely need to repeat the refrain when we move into physical science in the fall. For now, I’ll sing, shout, plead, or whisper in increasingly menacing tones: Show your work!

Review: Algebra Survival Guide

I recently posted a list of options for math beyond Singapore 6B. My younger, 10, finished that milestone a few months back, and I gave him some choice of what to pursue next. He selected The Algebra Survival Guide and The Algebra Survival Guide Workbook, understanding that they would not be a substitute for a full Algebra class but rather serve as an introduction. He agreed to that condition, so we began about two months ago. He was thrilled. I was satisfied. That’s about as good as it gets around here.

The Algebra Survival Guide, by Josh Rappaport, contains 12 chapters of largely pre-algebra topics. Broken down into bite-sized morsels, Rappaport explores mathematical properties, negative numbers, orders of operation, absolute value, exponents, radicals, and factoring. All those subjects are taught with variables and real numbers, but the real “algebra” part of the book doesn’t begin until halfway through the book, when he addresses factoring polynomials before moving on to canceling, equations, coordinate planes, and finally (though briefly) word problems.

For the most part, the Algebra Survival Guide breaks up those first concepts into page-long mini-lessons. Generally, the pages go beyond the “how to do this” and introduce why a property or process works. I like this. While there are times where memorization is a must, I’d rather math be deeply understood and utterly reproducible by one’s own mind and hand. Understanding how math works allows a person to do this. It’s a bit early to see if this understanding will stick,and he’s moderately mathematically intuitive, so I don’t know how much to attribute to the methods in the book, but I can say with certainty that this book does more than introduce rules to memorize.

Ironically, the book is also rule-heavy. In the process of breaking topics down into rather small parts, the author creates more rules than I recall from teaching my older son the same material in Jacobs’ Algebra.  In the section on negative numbers, these rules became burdensome, so we simply skipped those sections and moved on, after assuring he could do the problems themselves. The rules were actually a barrier to his intuition, so away they went. For a child struggling, these might be helpful and support understanding, but for my son, they got in the way.

What’s missing is the why of algebra. Until the final chapter on word problems, there is not a single example or explanation as to why anyone would bother moving all these numbers and variable around.  We stopped using the book near the end of the factoring section. I’d been growing restless with the teaching of technique in a vacuum, but he was progressing well and learning a good deal of the pre-algebra that Singapore Elementary Mathematics lacked (and saves for the secondary levels). Midway through a lesson on factoring polynomials, he asked the question: “Why would I do this?” With all the book had taught, there had not yet been one equation to solve, one word problem to ponder, or even one substitution of a number for a variable to consider. The “why” was missing.

I went to a bookshelf and pulled out Jacobs’ Algebra and searched for the section on factoring polynomials. We read through an example about a human cannonball’s trajectory. We talked for a while, and I realized that we needed to move back to math with context. He agreed readily, and we returned the Algebra Survival Guide to the shelf. Later that day, we ordered the first set of Singapore’s Discovering Mathematics series, per his request. He’s a creature of habit, and Singapore worked well for him. It’s worth a try.

I’m not sorry we spent the two months on the Algebra Survival Guide. It provided instruction on number of algebra and pre-algebra techniques with clear examples. It is designed not to be a full Algebra course but rather a support. It would serve quite well in this role. The text alone provides scant opportunity to practice the skills taught. Each one page lesson ends with four or five problems to solve, with the answers upside down just and inch or two away. Therefore, we used the Algebra Survival Guide Workbook for supplemental practice. For each page in the text book, the workbook offers ten to thirty problems for further practice. This was more than plenty, given the small bites in which the material was taught, but when we needed it, more problems were available. The workbook problems are rather cramped onto the page, with short lines for answers and no room for working solutions. This shortcoming was becoming more of an issue as he progressed through the book, and it does nothing encourage the student to show one’s work.   However, the book pairing was quite successful for what I desired as well – it served to introduce some topics missing from his knowledge bank in a palatable, gentle way. Mission accomplished.

On the positive side, the Algebra Survival Guide and workbook are easy understand, occasionally humorous, and fairly painless in their presentation of pre-algebra and the mechanics of working equations. They do incorporate the logic behind the mathematical concepts they introduce. They’re also inexpensive, with only the $10 workbook being consumable.

The chief drawback is the lack of context for learning algebra. Word problems make up the last chapter, but the approach is formulaic and is likely to do little to support a working understanding of algebra or help the user appreciate the skill they’ve learned much less an enjoyment for the beauty of mathematics.  Additionally, my 42-year-old eyes (which do not yet require reading glasses) found the font less than easy to read, especially the portions of small, fine print that explain why the various rules work. My son found my challenge amusing while I was just annoyed.

Would I use it again? Probably not. My son made great gains over these past two months, the largest being that he became comfortable with the idea of algebra. As I survey the other choices on our shelves and await the start of the secondary Singapore series, I know there are better choices out there — choices that support serious mathematical study while maintaining a humorous side. Ah, well. We have plenty of time to explore those materials while taking the next steps that Singapore has to offer.

Review: Jacobs Elementary Algebra

I wrote recently about options for math after Singapore 6B that we’ve tried or at least considered. While some of those resources found their way into my older son’s schedule while he was finishing Singapore, he felt strongly about immediately moving on to  ”real” algebra. He was nine and sick of arithmetic. He was also fascinated with the algebra I used to solve some of the more perplexing parts of Challenging Word Problems 6, Singapore’s last book in their honestly named supplement series. When I couldn’t make those bar diagrams work, I’d resort to methods more familiar to me. He wanted in on those methods.

After a moderate amount of research and consideration, I went with an old-standby, Elementary Algebra by Harold R. Jacobs (ISBN 0-7167-1047-1). Written in 1979, this black-and-white text is written with humor and interest without the distracting color splashes and sidebars that grace more modern textbooks. Perhaps those brighter, busier features and are a draw for some learners, but for my son (with ADHD), the less chaos on the page, the better. The cartoon at the start of most lessons held up well over those decades and grabbed my distractable child onto the page while giving us both a chuckle. A bit of a laugh is a fine way to start a math lesson.

There’s plenty of substance after that laugh. In seventeen chapters with four to nine lessons each, Jacobs takes a learner directly into the use of variables while teaching order of operations, graphing, exponents, radicals, and other pre-algebra topics not covered in Singapore’s first six books. For a mathematically geared child, this seamlessly integrates those missed topics into algebra, obviating the need for a separate pre-algebra course. For my older son who is highly mathematically intuitive, this was fine.

In Elementary Algebra, Jacobs does far more than teach the procedural goings-on of algebra. He explains why it works. This is not a text of algorithms to memorize and practice, practice, practice. Rather this is a book that encourages deeper understanding of the math it contains and that connects math to the greater world.  This creates a rather lengthy book, and my son did take a year and a half to move through it. At then end, however, he had a fine grasp of algebra and could easily relate and apply it to other studies.

The structure of the book makes for easy teaching, and the supplemental teacher’s guide (A Teacher’s Guide to Elementary Algebra  ISBN 0-7167-1075-7) provides additional ideas for teaching if that’s desired. This is, however, not a scripted program. For the parent whose algebra is more than a bit rusty, this text could be a challenge. Or, perhaps, it could be an opportunity to polish those rusty skills and dress them up with deeper understanding. Even if one doesn’t require the additional teaching tips in the guide, this book contains the answers to three of the four sets of problems in each chapter. (One set has its answers in the back of the textbook.) For this, it was worth its price several times over.

Each lesson takes a mathematical idea and develops it in two or three pages of text, diagrams, and examples. I’m a believer in interactive math lessons, since I think there’s much to be learned from discussion about mathematics. My son and I would sit together, with me reading the chapter aloud and discussing examples along the way, generally with scrap paper or a white board by our sides. Each lesson concludes with four problem sets: one review, two sections to practice the ideas from the current lesson, and a fourth presenting a challenging problem or two often with a historical bent or mathematical twist.  We generally omitted the review and did the second set (first set of practice problems) together. He’d then do the third set (second set of practice problems) and fourth set (challenge problems) on his own. The following day, we’d review his mistakes and move on to the next lesson.

Each chapter ends with two sets of review problems, of which I’d assign one. One review could be used for a test, but we used tests from the accompanying Test Masters for Elementary Algebra (ISBN 0-7167-1077-3), which offers four tests for each chapter, additional exercises on a host of topics, four multiple-choice midterms and two multiple-choice final exams. We’d have been fine without this supplement, but this was in my more obsessive “afraid we’ll miss something” homeschooling days. It’s definitely an optional supplement.

Algebra was more than a math class for my son. It was a jump in organization, textbook use, and test taking. Up until algebra, he’d done most of his mathematical work in his head. Dysgraphia and impatience with process had led to me scribing most of his work until this point, and while I’d modeled showing work, algebra was the first time I insisted he show his work every time. It was a painful first many months. The math came easily. Writing down steps did not. A second challenge presented when working through problem sets. Writing answers on paper while referring to a page in a book proved difficult. Often the writing issues, visual tracking work, and organized step-writing proved more challenging than the math. Test taking was also new to him. I don’t test my boys much — generally I can tell what they know and what they don’t. Test taking increased his accuracy and gave him a reason to show his work, since even a wrong answer with a clear and largely correct trail could earn partial credit.

Jacobs’ Elementary Algebra prepared my older well for the math that followed: Algebra II, Geometry, Trigonometry, and Precalculus flowed fairly easily from the lessons learned in that first algebra text. I enjoyed teaching from it, and he enjoyed learning from it. My understanding of some concepts deepened along the way. While it’s hardly the only algebra choice for the homeschooling family, Jacobs’ Elementary Algebra is a strong text based on sound pedagogy that prepares mathematical thinkers well for higher math.

What Do We Do After Singapore Math 6B?

What’s a parent to do after a child finishes Singapore Math 6B? After even the Challenging Word Problems 6 supplement? I’ve googled that question, asked other homeschooling moms on forums, and pondered it with friends each time a child neared that mathematical cliff.  There is, of course, no perfect answer. If your child is mathematically talented and young, delaying the algebra, geometry, algebra II, precalculus, calculus chain should at least be explored. (The Art of Problem Solving folks explain why rather well.) My older decided to start that chain immediately just before turning ten (albeit taking it at a slower pace), and he’s been largely happy with that choice. My younger son, ten,  more recently began life after Singapore 6B, so his path is still in its evolution.

So here are some possibilities. I’ve divided them up depending on whether we’ve made use of them at the pre-algebra/algebra level or not. Some are linked to my reviews while others remain on my ever-growing list of curriculum to review here.  Hopefully, the lists will provide some options for the child that gets to that point several years before his or her teens as well as for kids who reach it later. If you have more ideas, add them (with any links) to the comments section.

What We’ve Done

• Jacobs’ Elementary Algebra: When my older son was done with Singapore 6B at nine years old, he wanted to go right into algebra. Honestly, I didn’t know of any other path to offer, although his insistence was strong enough that I’d likely not have swayed him. Harold Jacob writes with humor, which appealed to my older son. Jacob quickly moves through the pre-algebra basics not covered in the Singapore series (negative numbers, exponents, absolute value, order of operations, and various mathematical properties) and onto a sound algebra program. (Review here.)
• Keys to Algebra: My younger used the first of these ten books while working on Singapore 5. The series is a consumable set of low-cost workbooks designed for a student to use alone. They are fairly dry but provide plenty of practice and start with pre-algebra concepts. I’ve known of families who have used these before a deeper study of algebra or along with the last books of Singapore.
• Algebra Survival Guide: This book breaks algebra down into bite-sized components, starting with a fair amount of pre-algebra (properties, exponents, negative numbers, radicals, etc.) Each page presents one concept, and many concepts are briefly proven along the way. For practice, there’s a consumable workbook with answers. While not a full algebra course in my opinion, it certainly does a fine job introducing many concepts that could make a more challenging algebra class more manageable. My younger is enjoying this book now.
• Penrose the Mathematical Cat: Theoni Pappas has a number of books for children and adults that introduce a variety of mathematical ideas that sit well beyond the traditional curriculum. My younger son and I started reading these together during the Singapore books and enjoyed discussions about fractals, mobius strips, infinity and more. These are definitely our antidote to math anxiety and traditional math blues.
• Life of Fred Prealgebra: Life of Fred has a growing body of followers. We’re not part of them. We tried the first book (reviewed here) and found it rather dull and lacking much math. If you’re looking for a light diversion for a few months, this may work for you. It didn’t hold interest here.
• Challenge Math: Edward Zaccaro presents an array of largely pre-algebra topics that are generally accessible before the end of Singapore 6B but could also be enjoyed after completion as a way of strengthening problem solving skills. His 19 chapters explore some pre-algebra basics (fractions, percents, volume, for example), trigonometry, algebra, and even a bit of calculus. With three levels of problems at the end of each chapter, there’s plenty to play with here. He has an algebra-specific title, Real Life Algebra, that sits on my Amazon wish list.
• The Number Devil: Fiction meets math in this playful trip through math topics that won’t be found in most math books. The author plays with combinatorics, fractals, Fibonacci numbers, roots and exponents, and more. A few folks on the web have created study guides and lesson plans for this book, which go a long way to flesh out this rather content-heavy book. Again, it could be appreciated before finishing Singapore 6B.
• Calculus By and For Young People: Here’s another book that takes meaty, real mathematical concepts and presents them in accessible ways for kids. While the author cites that kids as young as 7 could use it, it’s likely to be best appreciated with kids with a good understanding of fractions and decimals. It contains plenty of hands-on work with excellent explanations.
• Descartes Cove: Distributed by Johns Hopkins Center for Talented Youth, these six CDs make a quest out of math problems. The math is quite obvious and the pace is slow — it’s not a high-speed video game sort of experience. It’s aimed for middle schoolers and covers some algebra, geometry, measurement, probability, and more. The math demands higher-order thinking skills, putting it in a different league than the fast-twitch math fact games designed for the younger set. My older enjoyed this when he reached the last levels of Singapore.

What We’ve Not Done — Yet

• More Singapore Math: Two options exist for families that want to carry the Singapore experience through Algebra II. Both mix geometry, algebra, and other topics throughout four levels covering four years of math. I’ve held back, not wanting to commit to a program that makes moving to another more traditional program more logistically challenging. Both programs are said to be quite challenging. It’s still on the list of possible next steps.
• Art of Problem Solving: This publisher is known for rigor and deep thinking and is aimed at those that not only excel at mathematics (mine do) but really love it (mine don’t). Their pre-algebra book would be a fine place to start the series, which largely covers ground that Singapore 6 does not. Online classes (for a fee) and online problems (for free) could complement the text, although these books stand alone quite well. It’s still on my short list as well. My older tried the online Introduction to Probability and Counting course and was overwhelmed and discouraged. Like I said, it’s best for those students who love math, have ability, and really want to work hard.
• Thinkwell: This company offers a host of math courses from middle school to college. Instruction is via video, and problems can be done and graded online. Dr. Burger is an engaging instructor who adds enough humor to make the lecture interesting. My older son used Thinkwell for College Algebra (before they offered Algebra II) and found it appealing. However, for the child who needs a live human for instruction (and my older son does), this may not be the best choice.
• ALEKS: ALEKS is another online learning system. Unlike Thinkwell, it contains no lectures. Like Thinkwell, it generates and grades problems.  My older son used this for some of precalculus, but it has courses at the pre-algebra and beginning algebra level. Here’s my full review, but in summary, I recommend ALEKS as a supplement rather than a full curriculum.
• Khan Academy: This amazing free educational resource is a repository of chalkboard-style videos about math, science, economics, history, and more that’s growing by month. For many of the K-12 levels of math, practice problems follow lessons. We’ve used this for some cosmology, history, math, and chemistry over the years, largely as a supplement to other curriculum. He may not appeal to kids who need pizzaz to pay attention, but his methodical, clear presentation makes for fine watching for more patient viewers. Tracking  a student’s watching and practicing progress is available. My younger son has watched largely non-math topics, while my older son has worked through much of the precalculus lectures. Like ALEKS and Thinkwell, we’ve not used this for the pre-algebra or algebra level.

I’m sure I’ve left many post-Singapore 6B avenues left unexplored. If you have paths you’ve taken and would be willing to share, please add them to the comment section. Links to the sources are appreciated. It seems my next step is to review the resources we’ve used. That’s a bit larger job than I imagined when I set to making this list, which grew far longer than I imagined. I best get started.