Review of Kiss My Math and Math Doesn't Suck by Danica McKellar

Recently I had the delight of receiving a review copy of Math Doesn't Suck and Kiss My Math by Danica McKellar. Danica is well-known for her acting career, and her books are best-sellers.

First some basic info. Math Doesn't Suck covers middle school math topics: factors and multiples, fractions, decimals, percents, word problems, and a little algebra.

Kiss My Math covers pre-algebra: integers & negative numbers, variables and working with expressions, solving (linear) equations, word problems, intro to inequalities, exponents, and an intro to functions and graphing lines.

There's a website for each book: KissMyMath.com and MathDoesntSuck.com.

I've split this review into two parts: the pros and the cons. Let's go with the pros first.

Pros:

Danica's writing is excellent. It's conversational and catchy, very friendly. Reminds me of Dr. Math books.

I absolutely adore some of her illustrations that she uses for math concepts.

For example, she tells us to think about positive and negative numbers as breath mints - mint-egers. The positive ones have a good taste, and the negative ones are yucky, getting yuckier as they get more negative. And then she combines the integers as if they were mint-egers in her mouth! It's a fresh, fabulous comparison. You see, that'll stick to your mind for sure... you imagine TASTING something in your mouth, it will SURELY help kids remember.

Another thing I loved about her book were the little motivational "talks" in between the math stuff. For example, on p. 19 she asks, "Ugly Yourself Up?" Since you wouldn't make yourself to appear uglier than you really are, Danica argues, then why do the same as far as your smarts? She is truly encouraging girls to be smart, to study math, to do well in school and even venture into math careers!

It's really hard to put the book down, it's so delightfully written. How about remembering the order of operations by what Pandas Eat? And she goes further than the PEMDAS rule - Multiplication and Division are the main course, while Addition and Subtraction come at dessert.

For EVERY SINGLE topic she's come up with some kind of catchy illustration. For factoring, it's monkeys hanging off of trees. For the x (variable), it's a pearl bag, so 2x + 3 means "2 bags of pearls plus 3 loose pearls" (it's a GOOD illustration, I feel). For inequalities, she talks about how tall you'd want him to be? Perhaps h (his height) should be at least 2 inches taller but no more than 9... For isolating x in equations, she uses the analogy of gift wrapping and unwrapping. For exponents, there's the Ms. Exponent who's a high-powered executive. And so on.

You can see she's given each concept some thought, as to how to present it in a way that helps kids remember it.

She also helps us remember all these difficult math words (such as mean, median, mode, coefficient, variable etc.) with some clever mnemonics. I'll just tell you one. Mode is like "mood", whatever occurs most often, and she gives an example of a person who's iPod playlist contains mostly samba songs: "I guess she's in the mood for samba, huh?" Mode = mood.

The real power of the book is actually in these memorable analogies and in the very friendly tone she uses all through the book. She's talking to a friend, and it makes you to just keep reading!

Now, some might feel that it's all just nonsense. Will it really work to talk about Ms. Exponent being so powerful that all she has to do is say a word and things happen - you know, all she has to do is say "3", and 4 multiplies itself 3 times? Will it help students? Isn't it just silly to do math that way?

I feel it is truly powerful, because it has to do with our EMOTIONS. If it makes you laugh out of silliness, it creates an emotion while studying, and that helps you remember. It's a proven fact from brain research that we remember things better IF they're tied in with some strong emotion, be it fear, embarrassment, joy, surprise, etc.

You can also see from the feedback she's gotten that it IS working. Even math teachers have written to her how good her illustrative ideas are.

Another reason why this approach works is because she starts with the simplest stuff and in a friendly way "lures" you into the more difficult ideas. She basically "bends backwards" in order to not make math intimidating or scary - and that is one of her missions, as she has publicly declared.

I also liked the little "Watch Out!" boxes which contain alerts or explain common errors and misconceptions. Definitions come in "What's It Called?" boxes. Then in between the main text we also find "What's the Deal" boxes, Quick Notes, and Takeaway Tips, all decorated with smart and fashionable looking girl-figures.

Both books have some material on solving word problems. I feel Danica does an excellent job here and has really good common-sense advice AND great example solutions for this difficult topic. And I really liked chapter 12, The Art of Gift Wrapping/Solving Equations in Kiss My Math. It is one of the best "treatises" on solving linear equations that I've seen anywhere! The analogy of wrapping / unwrapping a gift is excellent.

Are the books just collections of tricks and shortcuts?

Some people have characterized Danica's Math Doesn't Suck book as simply a bag of tricks, concentrating on the "how-to" and not on the conceptual understanding. Well, it is written in a "how-to" style. It's like she's talking to you and coaching you through problems — and it's written so interestingly that it's hard to stop reading!

Sometimes she doesn't explain why things work the way they do, but most often she does include something along those lines - usually after the "how-to". So you can't say she's omitting the conceptual side (the why's). The books could use some more explanations on those lines, though.

Cons:

Now, I've checked around on the Internet, and it seems I may be a lone voice in the criticism that follows. However, I do not want to leave this part out so here goes.

You know, there are some of us (and I include myself in this) who don't feel we want our teenage daughters to start thinking about dating at such early ages as what her books audiences are. (Her books often refer to kissing, dating, having crushes, and so on.)

I'll give an example. She talks of integers as "mint-egers", which (like mints) taste like peppermint, spearmint, cinnamon and so on if they're positive. And they taste like vomit & dirt, etc. if they're negative. You combine (add) them in your mouth, and check which kind of taste wins. A great analogy! But right in the midst of several pages of discussing integers there's this one sentence: "These are good before a date, in case, you know, there's going to be kissing involved."

Another example: Chapter 9 in Kiss My Math is titled "Do You Like Him Like Him?" This catchy title is used as a mnemonic for the like terms in algebraic expressions — two terms such as 2x and 3x "get full-on crushes".


Now, she DOES also go against the popular culture in some ways. For example, she strongly encourages girls to study math, to be smart, to work at becoming smarter, not to feel that smart & pretty can't go together, not to underachieve and so on. The Kiss My Math book includes great testimonials from gals who overcame their struggles in math and are now "fabulously successful women".

I just wish she wouldn't go along with the "teen magazine culture" so much when it comes to crushes and boyfriends.

Recommendations

Teachers: I can heartily recommend these books for teachers, as they will be able to collect all kinds of ideas for their own teaching, and also learn from Danica's relaxed teaching style.

Parents and students: If you can keep from being influenced further into the boyfriend craze, go for it. If you can't, consider Dr. Math books (I've reviewed them as well). They are written in a somewhat similar, friendly, relaxed manner, though they don't contain nearly as catchy mnemonics and analogies as what Danica's do.

Review by Maria Miller

(The links below go to Amazon)
Math Doesn't Suck and Kiss My Math by Danica McKellar.

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You may leave comments on this blogpost, but only intelligent discussion, no flaming, please!

An AHA! abacus moment in the life of a preschooler

I've been doing math lessons with my 3-year old using the 100-bead abacus. Usually we do a few problems where she tells me a number to make and I tell her a number to make, back and forth. Today she asked me to make 51, and I asked her to make 68. These went smoothly since she's getting pretty good at this now.

Then we did a few "more than" problems. I said, "Let's say your sister has 5 cookies and you have one more than her. How many do you have?" This is a new concept to her so we need to do it slowly and carefully with the abacus: first make her sister's cookies, then let her have the same amount, then give her one more.

Then we do a few subtraction problems such as 7 − 4. She moved 7 beads, then "took away" or moved the other way 4 beads, and how many were left? 3 beads. I showed her also 50 − 10 = 40.

She started doing her own problem, "Let's do 9 − ..." and while she was thinking, I quickly proposed "... minus nine". Nine minus nine. Well, she moved nine beads, then COUNTED the beads one by one moving them the other direction, and was left with none... and what a SURPRISE it was to her! She had to start giggling!

I immediately showed her another one, 4 − 4. She did 10 − 10 herself. And THEN I showed her 100 − 100, which made the greatest giggles of all!

It was just so cute so I had to share. Plus, now you know several ways how to teach math concepts with the abacus.

Are these really parallelograms - answers

These are answers to my earlier post where I asked if certain figures necessarily are parallelograms.

The question was: Does the given information in each diagram guarantee that each is a parallelogram?

Figure 1:
This one you can't get around; it ends up being a parallelogram, actually a dandy rhombus. Let's prove it. You can notice it has lots of sides of the same length. If we draw a diagonal, we get two triangles with all kinds of same sides:

The two triangles ABD and BCD end up having all three sides the same. So by the SSS triangle congruence theorem, they are congruent triangles. Hence, their corresponding angles are the same.

I've marked the corresponding angles with the same colors. Actually the triangles are even isosceles so the blue and purple angles are even congruent... but we don't need that fact.

To prove ABCD is a parallelogram, we need to prove its two sides are parallel. And for that, it's often handy to use the corresponding angle theorem: if corresponding angles are equal, the lines are parallel. So image that we continue the line segment CD. Notice the additional green angle that I've marked:


How do I know it actually is a "green angle" (congruent with the other green angle)? It's because the three angles, being angles of a triangle, add up to 180:
+ + = 180. And the three angles being there along the same line (the continuation of CD), it must be. This sounds a little too complicated as I'm typing it. Perhaps I shouldn't have marked it green. Anyhow, since it IS congruent with the other green angle at C, then the line segments BC and AD must be parallel.

A similar argument would prove the other two sides parallel.

Figure 2: This isn't necessarily a parallelogram, but it IS always a trapezoid:



Figure 3:
This one is trickier, but it isn't necessarily a parallelogram. I used a compass to find a way to make this into a trapezoid:ABCD is a trapezoid with the non-parallel sides 5 units long.

Figure 4: This one is actually a repetition of the Figure 1, because it has the opposites sides of same length. We can use the identical argument to prove it is a parallelogram.

Multiplication vs. addition once more

Keith Devlin has published another column along the lines of multiplication not being repeated addition. I feel quite honored that he mentions THIS blog in his column (scroll down to the end), referring to what I wrote about the issue.

This time he expounds on research results. The research clearly shows that thinking of multiplication as repeated addition hinders students' further understanding of mathematics. It can lead to the misconception that multiplication always makes things bigger. Children need to acquire multiplicative reasoning, which is different from additive reasoning. And so on. Go read it yourself.

Bar diagram problem

This was asked of me as of today:

Please solve this using the bar/block diagram method. My friends and I are stumped....

Desmond had 480 more oranges than pears. After selling half of his oranges and half of his pears, he had four times as many oranges as pears left. Find the number of pears he had at first.
Thank you!

This problem is from a Primary School Leaving Examination (PSLE) paper. PSLE is the final examination for primary school students in Singapore. So, you would expect to see these kind of problems in Singapore Math.

My first attempt for solving this was like this:

It shows the difference being 480. The red lines are halving the quantities of pears and oranges.

But I quickly noticed this was way off. The amount of pears needed to be way less than the amount of oranges.

My second attempt was like this:
It was a little better, but half the pears looked more like 1/3 of the half the oranges. So the amount of pears needed to still be less.

This is the final diagram that solves the problem. The main idea is that the parts "match" - that you can see the relationship 1 to 4 in the amount of pears versus oranges, or in the halved amounts.

Actually, the CRUCIAL point of this problem - that there are 1/4 as many pears as there are oranges - can be understood without any kind of bar diagrams. You see, if that is true of the halved quantities (4x as many oranges as pears), then the same is true of the original quantities as well!

So without ever drawing anything, one can figure that there are 4 times as many oranges as pears. The DIFFERENCE is 480. The difference is also 3/4 of the oranges. So if 3/4 of the oranges is 480, then 1/4 of the oranges is 480 ÷ 3 = 160 (which is the number of pears). And the total number of oranges is then 4 x 160 = 640. That solves the problem then without the usage of bar diagrams or algebraic equations.

Bar diagrams can be of enormous help in visualizing the relationships between PARTS in the problem. BUT, to draw one is not necessarily easy because to get it right (like in my case) actually REQUIRES understanding something about the problem.

So, let the bar diagrams be a tool that helps you understand the problem ALSO in this sense: while you try to draw one, let the misdrawn ones guide your thinking towards the right ideas.

Are these really parallelograms?

Continuing with the idea in my post about Squares that aren't squares?, let's look at the following "parallelograms".

The question is the same:

Does the given information in each diagram guarantee that each is a parallelogram?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a parallelogram.

Figure 1:


Figure 2:


Figure 3:


Figure 4:


Again, these problems let students practice logical reasoning, and also learn about parallelograms, of course. See answers here.

Squares that aren't squares?

Updated with solutions!

Today I want to highlight a square problem I saw at MathNotations. I hope Dave Marain doesn't mind me showing this picture and problem on my blog... I have no problem acknowledging it's from his blog. I COULD just tell you all to "go read it at Dave's blog....

BUT I don't feel that's the best way, IF I want you to think about this. I can just guess that most of the folks would feel too lazy to click on the link and go read it there (would you?). So I want to show it here.


Figures not drawn to scale! And this is important!

Now here's the question:

Does the given information in each diagram guarantee that each is a square?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a square.

The IDEA is to make our students THINK LOGICALLY, or practice their deductive reasoning skills. A great little problem.

The answers:


Figure 1 is not necessarily a square. The upper left corner angle can be of any size. The upper side can be of any length. And so on. See here two examples.

Figure two is not necessarily a square either since the "top" side can be of any length. But it is a rectangle.

Figure 3 in the original problem IS always a square!

Now, I'll write another post where we'll extend this idea to some parallelograms.

Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the impression in children's minds that "multiplication reduces to addition" or, as Denise put it, it is a "subspecies of addition".

So, to try to summarize what I've been mulling over in my mind:

* Multiplication is simply a DIFFERENT operation from addition. It's not a "subspecies" of addition or some special kind of addition. AND, we need to stress that in our teaching.

Denise offers defining multiplication for kids as "counting by groups" - should be same-size groups, of course. In teaching that, we need to emphasize the meanings attached to the two factors: in M × N, M would be the amount of groups, and N would be the number of elements in one group. M is called the multiplier, and N is the multiplicand.

We need to emphasize the difference between additive and multiplicative situations in word problems:

"Mark has two baskets and each basket has five appples." => this is a multiplication situation
"Mark has five apples in one basket and five in another." => this is an addition situation.

Joshua says, "They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different."

It does sound simple and clear, right? I hope it does. The IDEAS are different even though the way we find the answer may be the same. But the IDEAS match these situations:

"Mark has four baskets and each basket has three appples." => this is a multiplication situation
"Mark has five apples in one basket and seven in another." => this is an addition situation.

Let's keep going.

• In multiplication, 1 is the special number so that if you multiply by it, "nothing" happens . Also called the identity element.
  
  In addition, zero has the similar role.

• Multiplication: each (real) number except zero has its multiplicative inverse so that if you multiply the number and its inverse, you get 1.
  Addition: Each (real) number has its additive inverse so that if you add the number and its inverse, you get 0.

• Multiplication has an opposite operation called division.
  Addition has an opposite operation called subtraction.

Pretty nice, eh? As we keep emphasizing these distinctions, hopefully we can develop in the students' minds the idea that they're different animals, not the same. One is not a special case of another. We can strive to define multiplication (initially) as "so many of the same-size groups" or counting by groups. Of course we have to FIND products by adding repeatedly, but we can treat them as different operations.

Then later, students will encounter multiplication of fractions and of decimals, and leave behind the idea that they solve multiplication by repeated addition. Yet, the basid properties of multiplication hold true:

• Its identity element is 1.


• Every number except zero has a multiplicative inverse (a.k.a. a reciprocal number)

I also found an interesting study quoted at Text Savvy. I quote:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . .

Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

Well, I'm trying. Here's the way I changed one page in my Multiplication book to read for now. What do you think?

Click on the link to download the one page (PDF).

Presentation at HOTM Conference

I just wanted to post a link to my presentation outline that I had yesterday morning at Heart of the Matter Virtual Homeschool Conference, if any of the participants (or non-participants) want to see it and see the links I gathered for it.

The Conference software didn't quite allow me to follow all those links during the talk so there were a few glitches. But I hope it was beneficial to those who listened. I was happy and excited for the opportunity to give the talk, though really nervous too! After all it was the first time for me to do such...

The idea of an online conference or virtual conference is just really neat! I was listening to the other speakers, and could get off any time to eat or do chores or whatever, then come back at my convenience. I got some ideas concerning scrapbooking your homeschool life, for example.

Icosahedron from picnic supplies

This was such a fun video to watch! This guy makes an icosahedron from plastic plates, and then from plastic cups - and one more from just plastic and duct tape.

It can serve as a fun summer math project for kids who love explorations, cutting, gluing, building, and that sort of stuff!

He is hilarious! See the video:


Video from Makezine.com

Read the instructions for the icosahedrons

Find out what is an icosahedron