Going Deep on the

# – Business Plan –

## Competition

### Ten competitors in the online algebra one course space

Competing products in the online algebra-one course space fall into three categories (print textbooks are excluded):

**Category 1. Online self-study courses.** The student takes fully-automated lessons or entire courses online. No live teacher is involved. AlgebraVictory! will be in this category.

**Category 2. Online academy courses.** A class of geographically diverse students are instructed live over the Internet by a teacher. Credit is typically offered via some accrediting organization.

**Category 3. Online tools for classroom use.** These are online support tools for use by classroom teachers that are typically sold ONLY to schools and are not available to homeschoolers.

Ten competitors are discussed in the following sections.

#### Competitor #1:

### Apex Learning

The Apex Learning platform teaches a lesson by presenting reading material, animated graphics, with voice over the graphics (you cannot see the teacher). The student can also download a pdf that contains a transcript of the voice and a description of the graphics. The platform then presents problems for the student to do in multiple-choice format. That is all well and good, but we must ask, What is the QUALITY of the lessons that get presented on this platform?

The Apex website offers no publicly accessible example lessons. One must create a login account (which we did) to see the actual content, which means we cannot provide any links here. Nevertheless, what we found is that the pedagogical quality of the Apex algebra one course is extremely low— *abysmally low.* We will critique the very first lesson, entitled “Rational and Irrational Numbers.” The very title of this first lesson suggests that the entire course is likely to be a pedagogical disaster, and a look inside the lesson itself confirms this! Why?

One of the bedrock principles of Mark’s teaching is this: **Do NOT attempt to teach students about extraordinary, seemingly impossible concepts, and expect them to answer questions about or do problems involving those concepts, without FIRST teaching that the concepts are extraordinary and facing the question of their very existence.** Apex lesson 1 is about rational and irrational numbers. The rational numbers are NOT extraordinary—they are simply fractions of whole numbers, whose existence is readily believed by most people. However, irrational numbers ARE extraordinary. It is by no means obvious that irrational numbers even exist, and showing students that they exist is HARD—it is definitely NOT an appropriate topic for lesson 1.

[Technical note: simply stating (as Apex does) that numbers with non-repeating decimals are irrational does not suffice until students understand both (a) that long division will ALWAYS terminate or produce a repeating decimal (this is not obvious); and (b) how to convert a repeating decimal into a fraction, which requires setting up two equations, subtracting them and then solving the resulting equation.]

In **AlgebraVictory!**, the existence of irrational numbers is first mentioned in Lesson 17, where Mark presents their existence as a “mysterious fact”—Mark delves more deeply into irrational numbers beginning in Lesson 57 with radicals, where he shows that the square root of 2 exists and proves it is not rational. In striking contrast, Apex lesson 1 ignores the fundamental question of whether (and why) irrational numbers exist, and then (reciting from the third item in the CCA1MS, “N.RN.3”) informs students that the sum of two rational numbers is rational, while the sum of a rational number and an irrational number is irrational. An absurd example will illustrate the pedagogical folly of this approach. Imagine that students are taught the following:

Boys and girls, today will learn about the different kinds of dogs. Some dogs have a head, and these are called headed dogs. Other dogs have no head, and these are called headless dogs. Now, if a headed female dog and a headed male dog have puppies, all the puppies will have heads. And if a headless female dog and a headed male dog have puppies, the the puppies will all be headless. Let’s take a quiz to see if you understand what we just learned. Suppose Fido is a male dog with a head, and Flippy is a female dog with no head. Will their puppies have heads?

**The students would laugh and not take such a lesson seriously!** Why? Because they already know enough about animals to disbelieve that headless dogs exist, much less are capable of reproduction! If the above lesson were to be legitimate, the first point the teacher should make is this: “You might think I’m crazy, but let me prove to you that some dogs have no heads.” That would get the rapt attention of every student, who would all be extremely skeptical. Students should likewise be skeptical about the existence of irrational numbers, but they do not yet know enough to be skeptical. Were Apex lesson 1 being taught live in a classroom, only a rare student would have the good sense (and courage) to challenge the Apex teacher by stating, “It appears to me that all numbers are rational. Why should I believe in irrational numbers? And why should I believe what you say about sums of rational and irrational numbers?” **Instead of learning the extraordinary truth about irrational numbers, student minds get filled with concepts they patently do not understand, and this spirals downward as the course proceeds. What students DO learn is that they do NOT understand the algebra.**

Moreover, even excusing the flagrant pedagogical blunder just described, the manner in which the Apex lessons are presented is extremely weak. It is basically a computerized presentation of the weak material appearing in so many algebra textbooks that have failed to teach the Middle 80%. And it is made even weaker by its close alignment with the nauseating presentation in the CCA1MS.

#### Competitor #2:

### Accelerate Education

Accelerate Education offers products in Categories 2 and 3, but does NOT offer any Category 1 self-study course. Nevertheless, an analysis of a promotional algebra one video from said course materials is relevant. It appears here (scroll down to the last small video at the bottom right, start it, then click the button for full screen mode). It is done with voice-over computer graphics. You cannot see the teacher. It includes snippets from the instructional videos.

It states that the instructional videos put a strong emphasis on somehow making the algebra “meaningful” and “relevant” to the students’ lives. Mark vehemently disagrees with this approach, which he finds is generally used to “prop up” an ineffective pedagogy. Students get bored BECAUSE it’s ineffective and hence, they are failing to learn the algebra. Then they ask, “Why do we need to learn this?” When algebra is taught well, students do NOT get bored—they become confident, feel empowered and realize that THEIR THINKING SKILLS ARE BECOMING MORE POWERFUL, and THAT is the reason for learning algebra! In a word, it’s “cool”!

The poor pedagogical quality of this course offering is revealed at 2:33 in the video, where the following image appears:

Displayed pursuant to the Fair Use Doctrine for purposes of criticism.

In an attempt to establish “motivation” or “relevance”, some text appears above the circle about organizing clothes into a dresser and sorting food items into a refrigerator. With the dresser/refrigerator example as background, it then talks about classifying the six numbers on the circle, but there is no mathematical reason for them being in a circle—some students will wonder what the purpose of the circle is and become confused. Is the circle like the dresser or like the refrigerator? The answer to this question is not provided in the video, but Mark provides it here: “NO. The circle is irrelevant. Forget about the circle. And also forget about the dresser and refrigerator.”

The circular diagram is attempting to distinguish natural, whole, integer, rational, irrational and real numbers. But these categories of numbers are NOT like the separate and distinct drawers in a dresser; rather, they overlap in interesting and subtle ways, like drawers within drawers. Indeed, ALL SIX numbers are real, which is not clear (and the circle certainly doesn’t help). One number is the square root of two, but the top is missing from the radical sign. Another is negative one, which is classified as an integer, and that is fine (although –1 is also rational and real, notwithstanding that two OTHER numbers have those respective labels). Unfortunately, *one of the expressions is outrageously inappropriate for algebra one students (and even for algebra two students)*: *e*^{i π}. This is the mysterious number *e* (which is not only irrational but transcendental and real) to the power of the mysterious number *i* (which is NOT real) times the mysterious number *π* (“pi”—also transcendental and real). This is a complex exponential. It’s not like 2^{3} which is 2x2x2, which is 8, as any sixth-grader can grasp. Rather, it is the bizarre REAL number “*e*” (which approximately equals 2.71828) raised to an IMAGINARY power!! What on Earth does that even mean?! Well, by Euler’s Identity, it turns out (even more mysteriously) to equal precisely negative one, and this amazing fact is far, far beyond the comprehension of algebra one students! (Ironically, Euler’s Identity is actually based on a circle: the unit circle in the complex plane, but that is patently NOT the circle that is shown!) Then, the chart classifies *e*^{i π} (which equals –1) as “real” *when negative one is already on the opposite side of the circle classified as an integer*! Thus, –1 appears TWICE on the circle, once in its usual notation as –1 classified as an “integer”; and again as *e*^{i π}, classified as “real”, thereby undermining (even for students who might somehow know the latter equals –1) the entire point of the chart, which is to show distinct examples of different categories of numbers. *This is atrocious pedagogy.* Since Accelerate Education showcases this on their website, it is likely indicative of the overall pedagogical quality of their course offering.

#### Competitor #3:

### Edumentum

Edumentum offers online products in all three categories, including “standards-aligned digital curricula for all 6–adult students,” including algebra one, that “Engage learners with online courses taught by your teacher of record.” (See here.) In discussing “Edmentum Courseware’s New Algebra Course,” their Senior Product Manager comes right out and acknowledges the Math Education Crisis by stating here: “Did you know that algebra is one of the most failed courses for high school students?” Yet NOTHING in their offering even remotely suggests their course can teach average students to do algebra. Instead, their algebra one course appears to be their latest rendition of educational establishment ineffectiveness.

The Edumentum website includes several screenshots of their algebra one course here. But the screenshots shown look like pages from yet another pedagogically disastrous algebra textbook. That same page states, “Students are also provided with detailed work-through videos which take them through the concepts step-by-step.” Yet, no sample videos are shown—if they had a good video or two, wouldn’t they showcase it? It appears likely that the videos are pedagogically ineffective and hence boring. Indeed, the page states, “Our algebra courses employ real-world examples whenever reasonable to provide a concrete context for the math and to convey clear relevance. This helps combat the age-old question from students, ‘Why do I need to learn this?'” However, Mark vehemently disagrees with this approach, which he finds is generally used to “prop up” an ineffective pedagogy. Students get bored BECAUSE it’s ineffective. When algebra is taught well, students do NOT get bored—they become confident, feel empowered and realize that THEIR THINKING SKILLS ARE BECOMING MORE POWERFUL, and THAT is the reason for learning algebra! In a word, it’s “cool”!

As stated here, the Edumentum algebra one course “is a completely re-designed course that offers 100% alignment to the Common Core State Standards for Mathematics. The specific standard alignment for each lesson is visible to both educators and students.” What this appears to mean that the course follows CCA1MS, item by item, which is a guaranteed pedagogical disaster.

#### Competitor #4:

### Glynlyon Teaching Technology (Odysseyware and Alpha Omega Publications)

Glynlyon is a Christian education company that owns two educational subsidiaries: Odysseyware and Alpha Omega Publications (AOP). Both offer online self-study courses (Category 1), where AOP’s offering is branded Ignitia. Both also offer online academies (Category 2), called Odysseyware Academy and Alpha Omega Academy. And AOP offers online tools for classroom use (Category 3) in private Christian schools.

To its credit, Glynlyon appears to distance itself from the Educational Establishment, and this is likely due to the longstanding anti-God, anti-religious stance of the establishment. Also to its credit, Glynlyon has a pretty good outline for its Ignitia Algebra One course here. We do not yet know whether Odysseyware uses the same outline.

However, based on this 1-minute example video linked to Youtube from Odysseyware’s website, the quality of the Odysseyware course videos is poor. It shows a lady facing the camera and standing next to a computer-generated graphics area. The graphics are small and difficult to read even in full-screen mode. She appears to be reading from a script, but her script is poorly coordinated with the graphics. For example, she begins by saying, “When **TWO variables** are directly proportional, their graph is **A straight line**…” In other words, **two variables and one line**, and that’s important because it takes two variables to make an equation for one line (unless the line is horizontal or vertical). However, as soon as she says that, after displaying a first line, a fraction of a second later the graphic displays a **second line**! That would be okay if she said, “Like this,” and then the first line was displayed, and then she said, “Or like this,” and then the second line displayed. She does not, which makes the graphics out of sync with what she is saying and confuses students. Moreover, although she moves her hands a little bit, she never draws anything meaningful in the air with them (like the slanted line she’s talking about), nor does she point to the changes that are happening in the graphics. It is unclear whether she is a teacher who understands the algebra or an actor who does not—indeed, the same lady appears in another video teaching English Literature here, and few teachers (including Mark) are skillful at teaching both subjects. Lastly, the lesson is trivial—it merely defines x- and y-intercepts, which are a very simple concept—it does not teach any significant concept or skill. Since Odysseyware showcases this video on its website, it seems unlikely the rest of their videos are any better.

We were unable to find any example of an Ignitia video, but AOP’s brochure here (scroll down to an image on the right side of a woman wearing a brown suit and orange blouse) shows a still image entitled “Ignitia Direct Instruction” that is strikingly similar to the low-quality Odysseyware video just described (although it features a different woman).

Thus, notwithstanding Glynlyon’s good algebra one-course outline, the quality of their Category 1 online course presentations appear to be weak. The instructional quality of their two online academy offerings (Category 2) will significantly depend on the teacher, but the Odysseyware video speaks “volumes” about what can likely be expected—if Glynlyon knew how to teach algebra to the Middle 80%, they would be showcasing that.

Lastly, nothing in Glynlyon’s course outline says anything about God being in the mathematics. Perhaps these Christians might not even KNOW that, much less know how to teach algebra in a way that invites students to sense the divine presence in the algebra. This may make **AlgebraVictory!** a better choice for Christians and other spiritual people who want their children to grow up experiencing God in their lives.

#### Competitor #5:

### Study.com

Study.com offers online products in Categories 1 and 2. Their self-study courses are for grades 6 through college, and each course consists of a series of short video lessons. Their academy is for college level only and has online teachers and proctored exams for college credit. However, the quality of their short algebra one videos is poor.

Their algebra one course outline is here and here. These pages list chapters and lessons, and each lesson has a short video that you can watch partially for free but then a paywall appears. However, if you stop about halfway, you can back up and watch another one. The videos show animated graphics with voiceover (you cannot see the teacher speaking).

There are several serious problems with these videos. First, the course outline is weak. Second, of the many we have viewed, each video only addresses a very short, simple problem. More complicated problems are ever addressed. Third, a chunk of time in each short video is devoted to irrelevant “motivational” material—teddy bears, ice cubes, rooftops, hamburgers, etc. that are intended to get students’ attention but have absolutely nothing to do with the points being made.

One of the videos here explains how to rationalize the denominator of a fraction that contains a radical, √ 7 , which is done by multiplying by √ 7 over √ 7 . That’s fine, but then she actually multiplies √ 7 time √ 7 to get √ 49 , and after that takes the square root of 49 to get 7. Yes that works, but it’s pedagogically weak for two reasons. First, students should already KNOW (by this time) that a square root times itself is the thing under the radical! Indeed, that is the very definition a square root. Going to 49 distracts students from understanding and recognizing this fundamental fact. Second, it is inefficient to multiply up to 49, and then turn around and go back down to the 7 again. What about the √ 791 times itself? Shall students first multiply 791*791 to get 625,681, and then attempt to find its square root? No, the answer is just 791.

The simple videos at Study.com do have some value, especially for students who need to brush up on something. But they do NOT comprise anything close to a complete algebra one course, much less one that can teach the Middle 80% to succeed at algebra.

#### Competitor #6:

### Khan Academy (Salman Khan)

Khan Academy offers online products in Category 1, in the form of short, FREE, self-help videos, plus minimal problem sets. (Notwithstanding the word “academy” in its name, there is no Category 2 offering.) Founded in 2008 by Salman Khan, who tutored his cousin over the Internet in math using a remote drawing program, the website now offers thousands of videos in a wide range of topics, including algebra one, here, which is taught by Khan himself.

The videos consist of Khan’s voice over a “digital blackboard” wherein he writes using some kind of digital drawing pad—what you see is a black background, his drawing cursor and the words and symbols as he draws them in various colors (and sometimes erases things). You never see his face or hands.

Salman Khan is an outstanding teacher, and his videos are GOOD.

Nevertheless, there are a number of crucial weaknesses that preclude effectively teaching algebra to the Middle 80%. First, as he admits here, he thinks the videos are “valuable, but I’d never say they somehow constitute a complete education. If I’m confused about something, hey, to get a 5-, 10-minute explanation of it, I think that’s valuable.” Indeed, his videos might help many students in the Middle 80%, but they are unlikely to succeed at algebra simply by watching these videos. Second, the course outline has numerous subtle weaknesses, especially regarding word problems.

For example, it appears that the very first word problem presented in the course is here. It states: “A factory makes toys that are sold for $10 a piece. The factory has 40 workers, and they each produce 25 toys per day. The factory is open 5 days a week. What is the total value of toys the factory produces in a day?” Kids in the Math-Smart 10% can figure it out in their heads—it’s just $10 times 40 times 25, which is $10,000, and the 5 days per week is irrelevant. However, most kids in the Middle 80% will be intimidated by this problem. Although Khan appropriately teaches its solution by the units cancellation method, Mark believes that method is better taught much later in the course AFTER teaching algebraic fractions. In contrast, when Mark teaches first lesson in word problems, he gives students a straightforward way to decode the words into equations. This builds confidence in the Middle 80%, whereas Khan’s approach leaves them wondering, “How do I figure out what to do with all these words and numbers?”

We congratulate Khan Academy for its significant contribution to algebra education. But **AlgebraVictory!** is MUCH better!

#### Competitor #7:

### CTC Math (Pat Murray)

The videos consist of Murray’s voice over a white background with computer-generated text, highlighting, circles, arrows, and occasional animated graphics. You never see his face or hands. Accessing the algebra lessons requires creating an account, which we did, but it means we cannot provide links here.

Pat Murray is an outstanding teacher, and his videos are GOOD.

Nevertheless, there are a number of crucial weaknesses that preclude effectively teaching algebra to the Middle 80%. For example, the very first algebra lesson is called “Order of Operations 1”. He begins it by presenting a rule called “BIDMAS”, which stands for Brackets, Indices, Division, Multiplication, Subtraction, Addition. Murray is Australian; in America, this is sometimes called “Please Excuse My Dear Aunt Sally” (PEMDAS), which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. He then teaches the application of the BIDMAS rule WITHOUT presenting ANY explanation of why the rule exists! The answer to the question “Why do we need or want this rule?” is both important and interesting—Mark NEVER introduces a rule without first presenting motivation for it, and this is important for teaching kids to think rationally about things, rather than just “doing as they are told.”

By contrast, in **AlgebraVictory!** Lesson 1 Topic 4, entitled Order of Addition and Multiplication, Mark uses blocks to show why mixing addition with addition is associative, why mixing multiplication with multiplication is also associative, but mixing addition with multiplication is NOT associative. Mark then explains that this non-associativity creates an ambiguity in notation that can be resolved by creating a rule for order of operations. THEN, with that fascinating background, he introduces the rule, but only as to addition and multiplication. As for subtraction, division, parentheses and exponents, those are addressed separately in later lessons, and the general scheme represented by BIDMAS or PEMDAS is taught in a much better way by syntactic analysis.

As a second example, in Murray’s first lesson on solving linear equations, he solves five example equations. The first two appropriately involve only whole-number x-terms and constant terms. Example 1 is too easy (because it has only three terms—four terms are needed to teach this with clarity so students can see both an x-term and a constant term get moved, respectively, to the opposite side of the equation). Example 2 is just fine—it has the pedagogical minimum of four terms. Then, rather than doing some equations with a larger number of x-terms and constant terms, which is the natural extension of what has just been taught (and then breaking for the student to do problems on that), in Example 3, he jumps forward conceptually by introducing parentheses into the equation; and in Example 4 he introduces fractions; and in Example 5 he introduces a fraction with a binomial on top! These concepts move WAY too quickly for the Middle 80%, who need practice at each of these steps, as well as exploring problems with more terms at each step.

We congratulate CTC Math for its significant contribution to algebra education. But **AlgebraVictory!** is MUCH better!

#### Competitor #8:

### Krista King on Udemy

Krista King on Udemy offers online math education products in Category 1, in the form of short, inexpensive, self-help videos, plus extensive problem sets. All are taught by Krista King, including “Become an Algebra Master” that covers algebra one and algebra two. Several free sample algebra videos are available here (scroll to the top of the page and then click “Preview this course” on the right), including an introduction by Krista showing her face and upper body. The free samples were insufficient, so we purchased a subscription for evaluation purposes, but that means we cannot provide specific links here.

The videos consist of King’s voice over a “digital blackboard” wherein she writes using some kind of digital drawing pad—what you see is a black background, her drawing cursor and the words and symbols she draws in various colors (and sometimes erases). You never see her face or hands (except in her introductory talks that have no blackboard).

Krista King is an outstanding teacher, and her videos are GOOD.

Nevertheless, there are a number of pedagogical weaknesses that preclude effectively teaching algebra to the Middle 80%. In particular, the course suffers from a somewhat erratic conceptual flow. The abstraction of variables is HARD for kids in the Middle 80%, and EACH new idea involving variables, ESPECIALLY combinations of variables, should be carefully taught in isolation from other concepts, accompanied by exercises. An example of this is adding and multiplying monomials, such as 3ab+4ab (= 7ab), or 3ab+4bc (on which one is stuck), or 3ab*4bc (= 12ab^{2}c and which involves exponents)—this comprises an entire lesson in AlgebraVictory. Yet, having not explicitly taught this (much less given students any practice on it), King throws these concepts into an early lesson in the course on the distributive property, with an example involving FIVE variables: mn(x+y+z). Then in the very next lesson, also on the distributive property, and having taught neither multiplication nor addition of fractions involving variables, she presents examples of the distributive property involving both! While creating complexity and confusion, this also distracts from the simplicity of the distributive property.

In a later lesson on solving a quadratic equation that has no real solutions (because the discriminate is negative), she ends up with √ –59 , which she changes to √ 59 times √ –1 , and then OUT OF THE BLUE she states, *as if it’s obvious and no big deal*, that √ –1 is the IMAGINARY number *i*, because *i*^{2} = –1. Say what?! HOW, ON EARTH, CAN A NUMBER TIMES ITSELF PRODUCE A NEGATIVE NUMBER?! This question should and will baffle kids not only in the Middle 80% but also in the Math-Smart 10%! One of the bedrock principles of Mark’s teaching is this: **Do NOT attempt to teach students about extraordinary, seemingly impossible concepts, and expect them to answer questions about or do problems involving those concepts, without FIRST teaching that the concepts are extraordinary and facing the question of their very existence.** The very existence of ANY number that, times itself, is –1 (or any other negative number) is nearly IMPOSSIBLE for students to believe. Simply asserting that this strange number *i* times itself is –1, and expecting students to believe this (and do problems with it), is outrageous. This is a HUGE algebraic topic that deserves extensive and careful treatment. Not only that, it is FASCINATING and MAGICAL—students LOVE to explore the concept of IMAGINARY numbers! Imaginary numbers are a major WOW!!! And their existence explains why the REAL numbers are called “real” (i.e. NOT “imaginary”).

We congratulate Krista King for her significant contribution to algebra education. But **AlgebraVictory!** is MUCH better!

#### Competitor #9:

### HSLDA Academy

HSLDA is the Home School Legal Defense Association, which provides legal services to defend homeschooling. It also operates an Academy that offers products in Category 2—students enroll in online classes taught by a staff teacher via a live online classroom interface, which includes a small video of the teacher’s face, a large digital whiteboard, and message areas for students to interact with the teacher (and perhaps with one another). One course offered is algebra one, and a brief sample lesson can be viewed on Youtube here (it’s best to put it in full-screen mode).

The teacher is Angela Ellis, and she’s a good teacher.

However, there are some weaknesses in her sample lesson. In it, she teaches solving an equation containing a variable under a radical: 5 = √ r – 3 . She receives live input and questions from her online students, including a suggestion to square the radical, which requires also squaring the 5 on the other side of the equation. She obtains the correct solution: r = 28. However, Ellis does not mention an important point: when you square both sides of the equation (OR when you multiply an equation through by an expression containing a variable), you can introduce false solutions—in AlgebraVictory!, Mark teaches students to write the word “CHECK” in the margin when doing this, which means each solution must be double-checked at the end, and any false ones eliminated. In Ellis’s example, since there is no variable on the left side of the equation and the 5 is positive, the squaring of both sides does not introduce a false solution; but if instead –5 were on the left, then the same solution (r=28) would be obtained, but it would be false.

At another point in the video, in response to a student’s remark, she writes the expression √ r – 3 + 25 . (It is unclear where this expression came from or how it relates to the original equation.) Apparently, the student wants to add 3+25 to get 28 (which ironically and confusingly is the solution to the original equation). She correctly states that you can’t add a number that’s inside the radical to one that’s outside it. However, both the question and her manner of answering it strongly suggest that the underlying syntactic structure of the algebra has NOT been taught. Were a student to ask Mark that question, he would react with surprise and ask, “What are you talking about?! Draw the syntax diagram and then tell me what you think.” Whereupon the student would quickly see that there is NO WAY you can add the 3 and the 25. This is not to slight Ellis in any way at all—Mark invented the teaching of algebra syntax diagramming, which is simply not offered in any other course because teachers to not (yet) know how to do it.

We strongly support the work of HSLDA in defending the inalienable legal right of parents to disconnect from the Education Establishment by homeschooling their kids. But **AlgebraVictory!** is MUCH better algebra course than their academy offers!

#### Competitor #10:

### Math without Borders (David Chandler)

Before discussing this competing algebra course, **we want to recognize and to honor David Chandler as a TRUE AMERICAN HERO**:

Math without Borders offers online math (and physics) education products in Category 1, comprising—according to its website here—a “Home Study Companion series [that] provides a complete high school math experience for homeschoolers by supplementing the best classic high school math textbooks… with solid teaching by an experienced teacher,” who is David Chandler.

The videos consist of Chandler’s voice over a “digital whiteboard” wherein he writes using some kind of digital drawing pad—what you see is a white background, his drawing cursor and the words and symbols he draws (and sometimes erases). You never see his face or hands. Students are required to purchase an algebra one textbook by Paul Foerster—each of Chandler’s videos is a lecture that teaches a corresponding lesson in Foerster’s book.

David Chandler is a first-rate teacher, and his videos are EXCELLENT.

Of all the competitors we have reviewed, this is clearly the best. We found no pedagogical weaknesses.

However, the videos are rather dry. You can watch two free sample Chandler videos here and here. His explanations are clear but he speaks (mostly) in a monotone voice. Undoubtedly, he loves to teach, cares deeply about his students, and is passionate about them learning algebra, but these emotional aspects do not come across well in the videos, which lack passion, excitement, and energy. That won’t matter for students who are strongly motivated to learn algebra, but kids in the Middle 80% do better with Mark’s lively, dynamic, passionate style of teaching (kids in the Math-Smart 10% also like Mark’s style).

Also, Mark disagrees with at least two curricular aspects of Foerster’s textbook. The first, which something Chandler *likes* about it, is the way word problems are presented. What Foerster does with word problems is far better than most textbooks, but Mark has an even better method. Mark believes students should be taught 15 specific kinds of word problems in algebra one, beginning with direct translation from his Word Problem Code. Chandler teaches a more difficult type of word problem here, which evidently is taken from Foerster’s book. Chandler and Foerster intend to teach students to think logically about, and apply algebraic concepts to, modeling real-world problems. Mark prefers to first teach them *mechanical methods* for doing the 15 kinds of word problems (which he spreads through the 75 AlgebraVictory Lessons), and THEN, when students have confidence with word problems, to teach modeling methods in GeometryPower and in second year algebra.

Second, Mark loves to teach the MAGIC in the algebra, and the crowning magical achievement in traditional first-year algebra is *the quadratic formula*. It is stunning and awesome—kids have never seen such a formula! Yes, it’s 100% logical, and yet it transcends logic (wherein the magic lies). And it is built upon other magic, including the existence of irrational numbers; and it portends later learning the magic of imaginary numbers. The quadratic formula also resolves a stubborn challenge—solving a quadratic equation that CANNOT be factored. Students should experience and face that challenge BEFORE learning about the quadratic formula that resolves it. Yet, Foerster presents the quadratic formula in the middle of the course BEFORE students have learned factoring polynomials, and hence BEFORE they have solved quadratic equations by factoring. This is like being shown how a magic trick works before seeing it performed—the wonder is totally lost! Ostensibly, this is to teach calculator skills, but Mark finds that a poor reason to sacrifice students experiencing the wonder. Calculator skills can be learned quickly later.

We congratulate David Chandler for his significant contributions both to education and to defeating the true enemies of America (and humanity). But **AlgebraVictory!** is a better algebra course!

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