# Wild About Math!Making Math fun and accessible

20Feb/0817

A while ago I discovered an interesting web site, Berkeley Science Books, that publishes a set of very comprehensive Ebooks called "Calculus Without Tears." Author Will Flannery has a pretty detailed explanation on the home page of his web-site of why he thinks Calculus can be taught in elementary school. His view is that Calculus in college is bogged down with lots of theory; if you change the focus of Calculus to application first and theory later, and if you teach the fundamentals of Calculus that don't require algebra, trigonometry, or geometry (except for the formula for the area of a rectangle) then you can teach Calculus to 4th graders. Flannery sees the motivation for all of mathematics, beyond basic arithmetic, to be physics, and the building basics - derivatives, integrals, and differential equations, which are fundamental to physics and to Calculus - can be taught to those with no mathematical sophistication.

Flannery questions the wisdom of the Math and science curriculum teaching algebra, geometry, and trigonometry before teaching the physics that drives the need for these other branches of mathematics. To be honest, part of me agrees with Flannery and part of me doesn't. I've always enjoyed pure and recreational Math. I absolutely love Math for the sake of doing Math. I love the logic, the creativity, the problem solving, the beauty, the joy, and the elegance of mathematics. But, I get that I'm not typical. Many people find Math to be too abstract and don't see the value of manipulating abstractions. For those people I can see the value of learning Math in a very concrete fashion. I can see the value in approaching Math from the desire to understand how our physical world works, starting with basic formulas for force and distance, and proceeding from there. I believe that someone with an engineering mindset or teachers who want to approach Math from the very concrete will really appreciate Flannery's books. I'm not an educator so I can't speak to what works best in the classroom. I would suspect that a combination of concrete and abstract might work best but I'm not sure in what combination or sequence.

I now provide a review of "Calculus Without Tears." My review is more about Flannery's approach to the subject matter, and not so much about the content, which you can read about on his web-site. You should know that I received review copies of the books from the author. Beyond that I received no payment of any kind and no conditions were placed on my review.

"Calculus Without Tears" comes in four volumes. I've seen the first three:

• Volume 1: Constant Velocity Motion, is 154 pages long and contains 13 lessons
• Volume 2: Newton's Apple, is 166 pages long and contains 14 lessons
• Volume 3: Nature's Favorite Functions, is 198 pages long and contains 18 lessons

Cost and ordering information is in the buy page.

The books are all in a workbook form. There are lessons with explanations of concepts and tons of problems per lesson. I really like this approach to learning. The exercises are so well structured that they guide the learner through the material one step at a time and thus build confidence. Thus, they are perfect for homeschool students or those who enjoy a self-paced approach to learning. I have to say, though, that while I believe that a motivated 4th grader could learn the material from these books, I'm not sure that a student with a 4th grade reading level would understand all of the explanatory material. Much of the material is easy to understand but I think some of it would require help from an adult teacher or tutor. An adult learner would sail through the explanatory material and benefit as much as a child from the numerous examples to work out.

True to Flannery's claim, very little mathematical background is required to learn the material. Volume 1, for example starts with calculating the position of a runner (running at constant velocity) against time. The student is given many opportunities to compute values of the distance function. First, values are entered into a table, later they are graphed. Derivatives are introduced as are integrals, all in a very natural progression from calculating values of simple linear functions, to learning how to plot these points, to learning about properties of lines, including their slopes and the areas underneath them.

Flannery employs a very clever technique to letting students know if they got the answers to the problems right. Many sets of problems have a "checksum", which is just a number that represents the total of the numbers of the answers of some or all of the problems in the set. So, a student computes the answer to the problems in the set, adds up those numbers, and compares it to the checksum. If his total matches the checksum then there's an excellent chance that he got them all right. If not, then one or more answers are incorrect and the student can check his work and change one or more answers until his total matches the checksum. While I like this approach I can also see some students becoming frustrated as there is no answer key and no solutions to the problems.

I appreciate Flannery's approach as an engineer to teaching this material. He has very clearly thought through all of the concepts he needs to convey to get from A to B and he has created numerous progressive exercises to cover that path. I'm quite impressed.

As a comparison of Flannery's approach to teaching Calculus to traditional methods, I pulled out my copy of Thomas/Finney Calculus and Analytic Geometry, which I used as a text during my senior year in high school when I took BC Calculus. Chapter 1, The Rate of Change of a Function, must have been meant as a review. It very quickly covered directions and quadrants, increments (deltas), slopes of lines, angles of inclinations, equations of parallel and perpendicular lines, point-slope equations, and general linear equations. And, that's just in the first 12 pages. Later in Chapter 1 is a discussion of limits, including limits involving infinity and those involving trigonometric functions. In my judgment, the Thomas/Finney approach requires a fair amount of sophistication and, as Flannery states, a solid understanding of algebra and trigonometry. Plus, Thomas/Finney is not relating their content to the physical world, i.e. their approach does not make the connection between Calculus and its motivation the way "Calculus Without Tears" does.

So, would I recommend the books? Yes. I really like the idea of demystifying Calculus, of having it be a natural part of early mathematics education. I don't like that Calculus is seen as something that requires enough sophistication that it can only be learned in an honors Calculus class in one's senior year in high school or in college. However, I don't see Calculus as fitting into a box where you either learn it early on or you learn it later. There's a place for college-level Calculus, and there's a place for getting grounded in the relationship between Calculus and the basic physics that motivates it. So, I would recommend that students get a hold of at least Flannery's first book and work through it to get grounded in the basics. If they resonate with Flannery's teaching approach then they can purchase additional volumes. Once students are grounded in the basics ala Flannery, when Calculus comes up again later in their academic career they'll have a perspective on what's motivating the material that will serve them regardless of how it's taught in high school or college.

1. I have these books and have just started using them, but have looked through them enough to be really impressed with how he introduces the foundations in a methodical and understandable, and yes, real-life format. I homeschool three boys, and my 10 and 9 yo, while pretty advanced in math (they love CSMP, and I’m using Art of Problem Solving textbooks with the eldest), aren’t quite ready for the standard pre-calculus and trig books. They are definitely ready for the ideas, however, and therein lies the challenge: how to introduce advanced concepts without having to use the advanced math notation. At the same time, I have always felt that the only reason I was able to do well in these subjects is that my high math teacher was, in retrospect, brilliant at introducing math concepts in a manner perfect for a visual-spatial kid (me). I always had a “picture” of what calculus concepts were trying to really do. I wanted to do the same for my boys (who are pretty V-S, too), starting with the slope of a line, which is a pretty fundamental building block. I searched through all of my math books, online, and these books were one of the few resources that fit the bill. Of course, the books go much farther than the slope of a line, which just saves me the effort of finding further resources!

I know that math theory, proofs, equations and so on are very important too, but the problem is that most math curricula are dominated with more abstract methods. It is finding the other kind (well-done and truly challenging for my boys) that can be hard. CSMP does both methods quite well, although they don’t use science principles and examples. I also like the AIMS books, some of the science books help to teach math quite effectively.

Sorry to be so verbose!

2. Kristine,

I appreciate your extensive comment so no need to apologize for being verbose.

I’m always delighted to hear from readers who have the real world experience homeschooling and teaching Math to their kids.

Thanks for the confirmation of the value of the “Calculus Without Tears” books. Seeing this approach to Calculus certainly go me thinking.

If you ever have a draw to writing a guest post on this blog on your experiences in homeschooling and Math I’d be delighted to publish them.

3. Hi Sol, thanks for the info and detailed review! I’ve had an itch to study calculus education, and this looks like a good resource.

I also believe the fundamental ideas of calculus can be understood by anyone (even a fourth grader), even if more complex things topics like computing the exact integral has to wait until higher math.

4. Someone who is thinking about teaching calculus to 4th graders has got to seriously consider the opportunity cost. If you are taking the time to teach calc to 4th graders, what are you dropping out of the curriculum to fit it in? This obviously depends on the individual student’s abilities and previous education. Typically 4th graders are still learning arithmetic and can be getting ready for algebra and geometry (if they are lucky and not in schools that teach mastery-free math). I haven’t seen the books you are talking about, but I would think it categorically a bad idea to forgo arithmetic, algebra, and plane geometry to get an early crack at calculus. There is no lasting benefit to enforced precocity. Also, the notion that physics is the real motivator for mathematics is completely crackpot. Physics and math are complementary, but come on, you can go deep and wide in mathematics (arithmetic through number theory, 4th grade through grad school) without knowing or needing any physics. My final question would be simply, what’s the point of learning calculus at a developmental stage when they aren’t able to master it (i.e. learn to compute integrals), given all the prior math that kids should know?

5. Hey, at last, a critic !

Well, we do compute integrals. We also cover the fundamental theorem of calculus. We even solve differential equations, which I claim are the sine qua non of calculus. All in the 1st book.

How do we do it? We restrict ourselves to studying constant velocity motion. Everything there is to know about constant velocity motion is in the formula distance = velocity*time (d=v*t). I did make sure my own 4th grader was good with this formula. That’s all the prior math you need.

We only integrate the velocity function. The area under a constant function is given by the formula for the area of a rectangle, the base is the time interval, the height is the constant velocity, the integral is v*t. I use the integral symbol for fun, but the calculation is v*t.

How about the FTOC, which I can phrase as follows: the area under a velocity curve equals distance traveled. As above, the area under the curve is v*t, so the FTOC is v*t = d . We already knew it !

DE’s? Here’s a DE, in words: a runner’s velocity is 5. The runner’s starting position is 3. Solve the DE (p(t) = 3 + 5*t)

So, you have to watch out for the language, there are big words in the book, differentiate, integrate, differential equation …. but the concepts are simple and the calculations always are d=v*t, or v=d/t. Here’s the dictionary:

Differentiate – calculate velocity
Integrate – find the area under a curve
Differential equation – velocity equation

It might be a good idea to eliminate the big words altogether.

Here’s the thing – I claim this approach really presents the fundamental ideas of calculus the way they should be presented, regardless of the age of the student. This simple minded approach is the way I understood it when I used it as an engineer. The rest is elaboration !

6. Thank you for this review! I was not aware of these books & am quite intrigued by the possibility of teaching calculus prior to algebra (& certainly trig!)

I do believe that calculus would have made more sense to me if it had been taught this way, and appreciate your comment that this is not an all or nothing proposition. I can see myself teaching calculus to the girls as part of their primary math studies, and then having them revisit it again later in high school or college.

Ruby

7. Thanks for the information and review of these books. I had no idea they existed! My DD is now homeschooled and absolutely loves math and science, which she didn’t like at her old school. With her uncle (and a couple cousins) being physicists, this will be the perfect introduction to the subject for a semi-mathphobic teacher like me. 😉

8. Any suggestions to teach a kinesthetic, dyslexic 6 y/o her multiplication tables?

Judith: Yes, I have some suggestions.

2. Buy Mark Wahl’s “Math for Humans” book. It specifically shows how to teach the Math facts to kinesthetic learners.

4. Subscribe to the Brain Integration blog it refers to, http://brainintegrationblog.com. I’ll get information about Brain Integration Technique and how it reverses dyslexia and post about it on that blog.

6. Most important – stop looking at dyslexia as an incurable condition. For 40 years I tried everything to cure my ADD. With time, persistence, and a gift from the universe I found Brain Integration and it cured it in 3 days.

10. I don’t have a direct suggestion for this question:

“Any suggestions to teach a kinesthetic, dyslexic 6 y/o her multiplication tables?”

but indirectly: why are you trying to teach a 6-year-old multiplication tables? At first I though this was randomly related to the calculus-for-4th graders, but there is the connection of trying to do things before (most) kids are ready. Is this really the time for times tables? For example, is she fluent with her basic sums and differences (you know, the addition table). If not, work on that.

For a direct suggestion, which I picked up from Saxon math, before teaching any row of the multiplication table, first learn to skip count . One typically starts with times 2 (2, 4, 6, 8, …) and times 5 (5, 10, 15, …) but before learning times-7, kids should know how to skip count by sevens (7, 14, 21, …) with ease. Sounds like common sense, but it seems that most strategies don’t do this.

11. Multiplication at age 6? Seems to early to me. But, let me add, for the elementary school math curriculum I’m a big fan of ExcelMath. It is to me the gold standard, absolutely great !

I also want to add to a claim I made in my previous post … I wrote “This simple minded approach is the way I understood it when I used it as an engineer.” and I knew that it sounds like hyperbole, but, in fact, it isn’t. So, I thought about that, and added a section to my web page titled …. ‘How Computers have Changed Calculus – By Making It Easy to Solve Every Calculus Problem using the Formula distance = velocity * time”. Yes, that sounds like hyperbole too, but I have an airplane simulator page that shows exactly how it’s done.

12. @bky: Thanks for your input. I’m not an educator and I don’t have a sense of what age kids normally learn things.

@Will: Thanks for the update and for your contribution toward making Calculus more accessible to everyone, young and old.

13. Multiplication at age 6? Zig Engelmann was able to teach algebra (as well as addition, subtraction, multiplication, and fractions) to 5 year olds.

http://kitchentablemath.blogspot.com/2007/03/engelmann-teaches-fractions-to.html

14. Funny, while this is seemingly late, being 2010. I was just wondering about this very concept. I have two boys of my own and as I get involved in their education, I have notices that we seem to be stuck in a educational rut.

It seems to me the ability to use logic/reason to determine the answer to a problem is more critical than memorizing that 23*6 is 138.

Basics of Calculus, the logic, seems to be far more valuable to education than memorization.

I will definitely look into these books for my kids.

15. Hello,

I think it is a great idea that Calculus is taught in 4th grade, because almost every job/career will involve some type of numbers so the earlier you teach them about math the better. My son just graduated from Junior High School and because of the math that was taught to him in elementary school and middle school when he starts high school now he will be at 11th grade math level, and he will just start 9th grade. This is why it is imperative that us parents teach our kids math at a very young age, because with us working with them, and the school system they can’t go wrong.

16. Part that caught my eye was using check sums to check answers. That’s cool, but when I learned calculus (I was doing it via correspondance course while I was in the army) I had the advantage of being able to check my answers at the back of the book. I could straight away see if I’d got it right or wrong and then go about doing it correctly. Once I started getting the answers right, I could then start doing a whole string of questions and get into a flow. That was when I really enjoyed doing math.

Personally I think that the faster you can check answers the faster that you can learn.

I think the argument for not being able to check answers is that people/students may cheat, but for those who want to do math for the sake of it then being able to check the answers is like learning tai ji and being able to ask the teacher straight away if you are doing it right or not.

On another note, re; dance of shiva, at one point doing long sequences I used the equivalent of check sums to check whether I was doing the sequence correctly or not.

17. I LOVE Mr. Flannery’s materials. I’ve been using since 2012 with minority students in elementary school, 4th and 5th grade. I start them off with Volume I, but I integrate it with a great deal of practical lab application using toy cars, trains and even foot races and race walking. The children easily understand these concepts. I am now launching these materials in Seattle area middle schools. I am deeply indebted to Mr. Flannery for his brilliant and well written materials.