I've always thought most math textbooks were a downright shame.  They don't explain things well, are usually boring, leave stuff out, and don't do much to challenge a student.  Besides that, most of them are used as a tool for teachers to be lazy.  Thus, I've always wanted to write my own.  I lack creativity unfortunately, but how awesome would it be to get some CREATIVE people to help out on the ordeal?

I do fear I'd blab too much however, and the book would be insanely long.  For example, I think one thing most people don't realize, is that math is not about NUMBERS.  I mean... sure, it uses numbers and calculations and formulas.... but I think the far more important aspect of mathematics is LOGIC. Math teaches you to reason through cause and effect, think critically, and make logical decisions.  Your thinking processes would not be where they are today without mathematics, and that's a very firm belief of mine.

What is one of the biggest mistakes a math teacher can make?  In my opinion, the most common error math teachers make.... is encouraging kids to -memorize- things.  I'm sure this sounds quite ludicrous coming from a teacher, but it's a huge issue to turn children into automatons that memorize something long enough for it to be useful and then dump it when it's not.  People only memorize things that are useful to them and let's face it... knowing how to solve a quadratic equation isn't useful to MOST people. When we as teachers don't focus on the child's understanding, we're only creating a snowball effect that damages a child's learning capabilities.

If, however, we can offer meaningful incentives, students might learn with more ease and with more purpose.  This is why I always try to emphasize to my students that math lass is NOT about numbers, but about processes.

I also prefer to ASK my student's questions.  There are many reasons behind this method, but one is simply to give the material more meaning. If I can invest a child into a conversation, I have them hooked and interested. Children get excited when they figure something out; it makes them feel smart and appreciated.  Thus, to me... being a teacher is about knowing the RIGHT questions to ask.  If I start off by asking a child "How do you solve a quadratic equation?", students will feel intimidated, stupid, and useless.  If, however, I start off by showing them an equation like "y=x²-4" and asking them how it's different from "y=x-4".....?  Well, then things will fare a lot better.  The idea is not only to alert students to their own abilities and build confidence, but also to model the correct pattern of self-questioning.  Most learning done in a lifetime, after all, is not done through the facilitation of a teacher.

This is the idea I want to keep in mind whenever/if ever I do write my own textbook.  The problem is the difficulty in holding a conversation with a reader through the text.  Is it possible?  Unfortunately, inspiring the reader to think creates a lot more hassle than many textbook authors think its worth, as the text must provide both question and answer.

As a small taste of how lengthy this can get, let me show you how I'd propose it's done through a simple concept.

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Division and Zero

Essentially, any fraction represents one amount being divided by another.  For example, if I were to show you the faction 24/2, this represent 24 'items' being split into two equal groups.  This fraction could represent anything from 24 cookies being split between two people, to 24 hours being split into two day halves.

Let's then look at the following scenarios. One of my classes contains 23 kids.  If I had 46 pieces of candy and wanted to split them between my students, I would represent the situation with the fraction 46/23. This means that each kid would receive two pieces of candy, because 46 items are being split between 23 students.

What if instead, I asked you to describe the faction 0/23?  This would mean that I had 0 pieces of candy to split between 23 students. No matter HOW badly I wanted to give my students candy, I would run out of items to split before I had even began to pass them out!

Both of these fractions make sense and result in a definite answer.

On the other hand, consider 46/0.  This would mean that I have 46 piece of candy, and 0 people I wanted to split them between.  I could walk up to a student, show her the candy as though I'm about to hand it to her, and then pull my hand back with a laugh.  "Ha!  Just kidding!" I could joke.  Now here's the interesting question?  How many times could I pull this little prank?

If you answered 23 times... one time for each student, think again!  I could walk up to my first student and put on my little act... "Ha! Just kidding!"  As long as I'm sure she won't attack me in fits of rage... couldn't I do it again?  I will still have 46 pieces of candy no matter -how- many times I pull the joke.  Therefore... ideally... how many times could I pretend to pass out candy but keep it for myself?

26 times? 100 times?  No.. I could do this forever as long as I had breath to live.  And as long as I wasn't limited by time?  I could do this an infinite amount of times.

So why isn't 46/0 equal to infinity?

Well, remember the other two situations?  46/23 was equal to 2, and 0/23 was equal to 0.  In fact, any fraction we look at has a value.  Even 22/7 gives us a numerical answer (3.14159265358979...), and each of these results is very -definite-. To say that 46/0 is infinity is quite the opposite.  Can anyone contain an infinite number of objects?  Could you hold an infinity specks of dust, or could a pool hold an infinite number of gallons?  The answer is quite obviously no.  Thus, any value divided by 0 is undefined.

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Now, how do most books describe this concept?  Simple:  Any number divided by 0 is undefined.

Any textbook which even mentions this fact, if you're lucky enough to find it, says no more on the matter.  When students see this statement, they picture a few images in their mind: fraction, 0, undefined = no answer.

Then when they go to work a problem and see something like 0/5, they become excited and proclaim "The answer is undefined!"  The problem has no meaning to them, and will always be a point of confusion, with little opportunity to fix the mistake.  Why is something divided by zero undefined?  What does undefined mean, and what does it matter?

Unfortunately, going my route also means a longer textbook that's bound to cause fear in our students.  As sad as the fact might be however, there also are not enough teachers willing to explain things beyond the scope of children's textbooks. What's the compromise?
 What are some other hits and misses in the textbook world, and what do you think can be done to fix the matter?  Do you agree with my opinion, or is it too much hassle?

Some people I'd like to hear from if they have the time:
rcarter2 :
play4fun :
Light Knight :
kramer4077 :
KiyokoNee :

I know some of you are on different levels and might not even use the same types of textbooks, but I'd still like to know your professional opinions as people invested in education.  

I strongly agree with your idea. Coming from person like me, a straight F student in math. A large textbook would be scary. But if introduced right, could be very great for the students. Its either you interact a lot with the student or have a good textbook. Or if you're lucky, both! I had to go with a teacher with 0% interaction. With a terrible textbook I didn't do so hot along with many other students.

Honestly you might want to be more interested in students' opinions, as well as teachers.  Like you said, is the textbook a tool for the teacher to be lazy, or a tool for the students to learn.  Well that would mean your target audience shouldn't be the teachers, but the students.

Hopefully (and with the way our education system works sometimes, I wonder) you could assume the teachers already knew the knowledge in the textbook and that if the textbook was geared toward a student, that student might actually read the book and learn from it.  I think that boring textbooks are a problem as well.  I can remember a few times in school when I actually did get into a textbook and wouldn't stop at the assigned place just b/c wanted to keep reading.  If more books had accomplished this more times I would have learned more.

Now with Math it does seem a more difficult idea to implement.  Science seems to naturally draw the curiosity of children, History can be romanticized.  What do you do with numbers to make them fun or at least interesting?

Well I'm gonna look at it a little like say, teaching English.  The letters and grammar  themselves are not really interesting (to most anyways), but a story certainly is.  So while the numbers themselves are not very fun to learn, the concepts behind them can be.

I want to address a statement that you made at least twice which is that the textbook would be too large. First of all, a large textbook may not be a problem if  more and more schoolchildren are learning through mobile devices. In ten years a document like that might be applicable. Also, who ever said that there couldn't be topic specific math books. Why does a textbook have to have so many chapters? Why can't each chapter be it's own resource?

If you had the heart for it, you could write teen fiction where the lessons are incorporated in some way. 

thephantombrain : Interestingly enough, I was actually going to mention something you said last night, but I chose not to because I didn't know how many people would think I was absolutely insane.  I was actually -literally- going to ask: "Why can't we just have a textbook for each individual topics?  Or a few topics?" 

You're right about technology though. I actually never considered that.  We ARE trending towards a paperless society. Even many schools are giving iPads to their kids, so that their kids can keep all their textbooks on the iPad.  Maybe now is my time to get rambling after all! haha!

fightorace : I completely agree with you on the issue about hearing from students as well. To be honest, I DID want to hear from students.  I mainly mentioned the 'teachers mainly' portion because I was afraid of this thread being filled with TdV spam. I wanted thoughtful responses, not ones designed to increase post count.  I figured adults would be the ones to not concern themselves and actually give responses like I was looking for.

Textbooks can only do so much.  Children need the opportunity to ask questions to someone they trust, and get answers from them, and perhaps more often, questions.  If the children answer the questions using their own abilities while having guidance from their teacher, they can have something they can be proud of.  Too many children fail to take this initiative, though.  The teacher can be looked upon as faulty for the mistakes of these children, but teachers can only do so much.  Children cannot be made to learn, but they can be taught how to learn, so that in the future, if they so choose, they can take the initiative and figure out the whys and hows on their own using the knowledge they already have.  Critical thinking is so vital these days, and is missing from too many media sources, including TV shows and video games.  Too many video games reward the player for fast reflexes, rather than a trained mind.  Games such as Crashmo and the latest Zelda game have succeeded, though.  Crashmo encourages players to think outside the box to solve puzzles.  The Legend of Zelda: Skyward Sword encourages BOTH critical thinking and fast reflexes, and requires that the player use both mind and body together to solve problems.

I think textbooks are only to be used as an assisting tool, not the source of a lesson. When I was teaching middle school science last year, I had to cover chemistry, earth science, and space science. 2 of those were topics I had not taken classes for in over 2.5-3 years (since they aren't biology). I was super tempted to rely on those textbooks. But I ended up having them read those textbooks about 3 times for the whole semester. I find that textbooks are good when there is a lot of terminology to know. It can be an effective way to introduce them to that terminology so you can get them some prior knowledge to activate later. We benefited way more from experimenting, and trying to logically come to conclusions as to why something happened the way it happened. Then we would discuss our ideas. I would then give them rundowns of how things work, and then we applied it to what they did in the experimented to see if their explanations changed (which most of the time, they did). 

In my current theater I class, we have not used a single textbook. I have not used anything close to a textbook in stagecraft. All that is best learned with firsthand experience that you can't get from a textbook. But in a theater class, that is just how it is. Textbooks can be VERY important tools. But they need to be used as just that. Tools. Not a source. The teacher is the keeper of the knowledge. But teachers aren't supposed to just give the explanations. They need to be the keeper of the knowledge so they can be the facilitator of the student driven learning, where they might use a textbook as a reference only.

rcarter2 :  I completely agree with you on the fact that nobody should RELY on textbooks. In fact, my kids really don't have to bring their textbooks to class because I don't teach out of one and never have. Students usually use the textbooks from home as an extra resource.  My main concern is that (in math especially), I don't feel as though they are GOOD resources.  Students don't take good notes and hardly seem to remember what their teachers say. I just wish their was a good compromise into getting past that barrier.  I'm certainly not saying that a child should learn everything from a book!

I prefer my kids to experiment and learn.

However, if they go home and wish to look up 'dividing by zero' because they can't remember what I said or didn't take good notes, how is the typical textbook going to help them? The answer is simple: it won't.  This is why I think they need some sort of revamp.

Singelli : One huge problem is that this generation is one that has to get information instantly. With mobile devices, Google'ing, etc. kid's attention span is dwindling. All the things they deem important are things that can be looked up instantly. So their attention spans are geared towards instant. More advance concepts in school (like math) are not something you can get instantly. It requires real critical thinking and requires you to not only read, but REALLY read. Internet reading has adapted kids to just skimming through things quickly only looking for specific cue things. They can hit Ctrl+F to find key words instead of reading to find what they need. You can't get math instantly, and it requires time and work. We are putting ourselves in a situation where if it isn't something that is instantly gotten, it isn't worth doing. People rely on calculators to do their work. But the problem is that calculators are made by people who understand how the math works. Where are we going to get our calculators when the whole generation never learned the math because calculators did if for them? (Calculators is not literal here, more of a metaphor in that the new generation can't continue the current technology, let alone advance it, when they don't know how it works because technology has taken that knowledge requirement away).

As an instructor, I agree with your idea, and with the addition of technology to the mix, we have a very viable prospect. Perhaps the start would be to encourage more teachers to explain the "whys" and "hows" of a particular topic, or to have more interactive settings to at least make it interesting. Many students have a harder time focusing on a topic in which they don't understand, and are just even more likely to avoid asking for help if this is the case. Of course, explanation comes with a bit of a balancing act, but it can be done. As a student, if the problems are understood to be in a certain context, such as programming software to make a robot move, or actual measurements made to adjust the wiring in your house, I find the concepts much easier to grasp and much more solid in terms of how long I remember or understand them.

I suppose what I wish to see is for teachers to create an environment close to something that happens, such as where divide by zero occurs in terms of jobs. It doesn't have to be the "this happens here, there and over there" type of circumstance, but at least a small example to provide students for the awareness of why these topics are covered at all. They wouldn't exist if no one had a reason to look into them.

Singelli : I keep meaning to respond to you as well as take a photo for you of our new third grade math workbook.  I haven't had a chance to take the photo yet, but I will respond anyway   We are using MyMath by McGraw Hill and each lesson has 2 teacher instructed pages, 2 independent student pages, and 2 homework pages!  It's absolutely craziness the amount of work that's included in this workbook!  Workbook 1 is over 500 pages and workbook 2 must be about 300!  There are many good things about it though, it comes with an interactive website
that teachers and students can log on to.  There are lots of games, songs, and resources for teachers on the website.  I also like that if a student is absent (and they are smart enough) they can oftentimes just look at the teacher-instructed pages in the book and teach themselves what they missed.  This new series seems to scaffold information at a good pace too.

Our old math series was called Everyday Math and that had a little hardcover textbook with it called the "Student Reference Book."  I kept them in my room, but with so many pages in our workbook, there is no time for the activities in there!  Those books were great for allowing students to access a variety of information.  However, third graders are a bit young to seek out information on their own and usually needed assistance when using the textbook.

I think that workbooks and textbooks should provide students with enough information so they can get caught up on a lesson if they are absent, or as you said if they forget something and need to look back at it later on.  However, the thickness of our third grade workbook is just INSANE and I will have to take a picture of it for you next to a regular-sized notebook or something to compare it to!  They kids hate hauling it out of their desks, but it does include everything we need to talk about during a lesson.  Yet, our workbooks for our old series Everyday Math were lacking in so many areas (like practice with basic facts).  It seems like there should be a happy medium somewhere with a more concise textbook or workbook!

I always thought it would be so fun to be one of those big wigs that develops textbooks and curriculum.  I already have my Masters in Curriculum and Instruction, so maybe you and I should go into the textbook-writing business, Singelli

kramer4077: That workbook sounds absolutely amazing, but doesn't it intimidate the kids and their parents?  I find it pretty interesting that your workbooks sound so thorough while ours are anything but. The problems in our students workbooks are nearly pointless, and there are literally no explanations in them.  There is a separate 'workbook' called a note taking guide, but it's a simple fill-in-the-blank novel really, giving students very few example problems and hardly requiring the students to think.

I actually discovered this summer that I greatly enjoy lesson planning of its own accord.  However, the way our system wants us to do it is absolutely ludicrous, and on top of the work we already have.... nearly impossible.  I actually wanted to ask around earlier this summer to see if it was possible to make a decent salary by solely writing lesson plans for a state maintained website which serves as a collection of resources for teachers. I've ALWAYS wanted to write a textbook too, and my students always tell me I should.

It's just a huge task to take on between everything else.

KiyokoNee :  I am totally with you on your 'application' ideas.  Students NEED to know why the stuff they learn is important.  However, it does get more and more difficult with math. Students are getting smarter in the sense that they know how dependent the world is on technology.  More and more often lately, I don't get the question "Why do we have to learn this?" but instead "Why do I have to learn this if there are machines that will do it for me?"  They understand that machines are fallible, but they don't really care, either.  With the reliability of computers and technology, can we blame them?

This is a huge reason I've been struggling more and more with my dislike of calculator dependence.  I believe all students should be able to perform simple mental calculations, but when they can punch something into their cell phone and get an answer seconds later..... and when technology is nearly always present in their lives...... how upset can we get?  It breaks my heart when I grade papers and get stuff like 3-1= -4, but then again.... I understand the kids have been coddled.

But let me step away from that rant.

Wouldn't it be interesting if we stepped away from textbooks and instead gave the kids video series in place of a thick book?  I think in math especially, it would be more beneficial...  I wonder why we haven't done that already.

rcarter2 :  Since you kind of touched on my small rant above... I thought I'd summon you back to this discussion.