why would the teachers not think the books are a good idea? memorising formulas will only get you so far you need to learn problem solving skills think logically but also if everything is in a context would it not make students think what is the use of this when maths isnt just applied to use maths is also theoretical
Also, nice example of dividing by zero in the first part. The way I always thought of it was that you can't take something and divide it among zero people. You can divide it for one person by leaving it as a whole, but not among zero people.
@jamesblackburnlynch Hmm, never really thought about that one before. I was going to say that you can't divide anything among zero people in the first place because, if you were going to divide anything (even if its amount is 0) among anyone, there would have to be at least one person, but I understand what you're saying there (like if you have nothing to divide). And I guess it doesn't really work as well as what you said because you can divide by a number smaller than 1.
@scoobertYTP If you think about it the way I described for 1/0...that is, 0 times what equals 1, then for 0/0 the question becomes 0 times what equals 0? And the answer this time is...every number there is. Anything times 0 will give you 0. (Which matches the idea that if you 0 things you can divide them up among 0 people anyway you want...since there are 0 things. Sort of. Some of the intuitive nature of the 1/0 as breaking up into groups is lost.) Too many answers to define!
@jamesblackburnlynch Interesting thing you're pointing out there. If you think about it that way, anything times 0 does give you 0. Yet you can't divide by 0. I sometimes think about it was as if you actually had to give something (or nothing) to someone by dividing. If you tried to give it to zero people (including yourself, which would be 1) then you couldn't do it. And it sort of works for numbers between 0 and 1 because you're multiplying instead. But what about negative numbers?
@scoobertYTP I can't pretend to understand what you are saying there. Division really is what I'm saying. I'm not coming up with anything clever. 6/2 means "2 times what =6?" So 1/0 means "0 times what =1?" As we've said, that's impossible, so 1/0 is undefined. And 0/0 means "0 times what =0?" That's everything. Since mathematicians want a unique answer to every question, and no answer is any better than any other here, we choose not to define 0/0. That's all.
@jamesblackburnlynch Eh, the thing I'm saying is extremely complicated and is very hard to explain, especially since it doesn't really work for explaining negative numbers. For some reason it always just worked for me. But I agree completely with what you're saying division is. And now I understand completely what you mean with why 0/0 is undefined. There are infinitely many solutions to 0 times what is zero, which means there are too many solutions to define one.
@scoobertYTP Well, that's exactly what I mean, so yep, what you are saying is probably just too hard for me to understand in 400 whatever characters. Now...why is 2^0=1? Enjoy...
I agree about your ideas on what math should be. Up until about a year before I started college, I was one of those people who thought that math was just a bunch of formulas that really didn't relate too much to life. After that I realized that, if you don't just do it by memorizing formulas, it causes you to think in a certain way and helps you apply concepts involving logic that could be valuable in the future. I'm better at math now, but I still wish I would've realized this sooner.
You just confirmed her argument with your closing statement. Everyday Math is concerned with understanding math ideas WITHOUT spending time to master them. So in essence, there is a problem with those books.
@gothamvengeance Actually, her argument has more to it than you think. She is arguing about the focus of math education. She wants it to be on mastering efficient skills. It's true that she wants them to have more time practicing, but she also wants students to be shown what is the "best" way without wasting time on less efficient methods. That is, she sees math as a collection of skills to be mastered. I don't. But I do agree, there's nothing wrong with mastery of skills.
@jamesblackburnlynch I appreciate your response. From my understanding, the Singapore method (which is the subject of the books she recommends at the end of her video) has a balanced focus on understanding the ideas *and* mastering the skills.
I am a parent of a 5th grade daughter who is struggling mightily with the Everyday Math program. I agree that basic critical thinking should be a goal of what is being taught, but not without the mastery of doing so.
@gothamvengeance I might have to watch her again, but I believe the books she recommends are the Saxon books, which I do not like at all. They are all about skill practice.
Singapore, on the other hand, I'm a big fan of. It does both the conceptual and skill sides that I want. Over the years, I've come to the conclusion that EM is just too challenging. I think one can supplement the skill mastery, no problem, but there is too much in those books.
I find the traditional methods simpler to understand and to explain to young children. If properly taught they make sense. There is no more intuition or logic to any of the newer methods I’ve seen. The reasons certain people have issues with math are many and complex. I don’t think properly teaching and explaining the traditional algorithms is one of them.
I have advanced degrees in Math and Statistics and have taught at the college level. I do not think anyone is saying you should just teach algorithms in a vacuum with no explanation as to why they work. If explained properly the traditional methods have just as much, if not more, logic than those of the other methods. I have seen this taught properly to my 4th grade son using diagrams and word problems to show their relevance.
A+ I teach Everyday math at my school. Many teachers struggle to teach it because they don’t truly understand math themselves, which of course leads parents to believe there is a major problem with the curriculum themselves. They try to teach it as rote memory (like “old” math). I’m a math person and developed my own strategies as a child. Non-math people never create these strategies and end up hating math. EDM tries to teach all students what math people have been doing for years.
The problem I've seen with the new math texts is that they often preclude parents (who learned traditional math) from being able to step in and help when their child is having problems. What happens is the parent is confronted with a jumble of terms and methods that they aren't familiar with. Then the parent gets angry because they can't help their child.
This problem will presumably disappear in a generation. But I think it could have been avoided if more thought had been given to this.
When I was learning math, the key to my success was just that. Context, algorithm, theory, and practice. Solving problems in different ways was also important. Versatility.
I really appreciate this video response. I am a college student with a minor in physics, and as I was listening to "An Inconvenient Truth," I started to feel more and more like the claims in those videos were unjustified. I'm glad to see someone with a real sense of what mathematics is and what mathematics can achieve speaking to the issue.
Thanks for your response, by the way what the lady called "standard" it is here, I learn a similar but different way to multiply, that she is more proficient in one way or another that does not mean it is wrong. It is as you say, much better than me, another way to see it to understand better of what is behind the formula
I find the idea that the weather forecaster presents - which is there is a vessel (student) into which a method is poured - frightening in its ideology - that isn't how education works. The students are not empty vessels and the teacher is not pouring some pure knowledge into the child's head.
Thank you so much. I am also a college math teacher (after 7 yrs as a h.s. math teacher). This is my first experience with YouTube and I searched on Math first. My daughters successfully learned math with Everyday Math and both completed AP Calculus with 5's. As you said here, people have been saying the same things about math proficiency and education in general for decades (actually, centuries). It's interesting that the original video was not done by a math teacher but a meterologist.
What a passionate and enlightened dialogue. If only we could have these kinds of real conversations about all of our problems in the education system (math, the lack of civics, etc...) and the problems in our country generally, we would be a lot better off!
Ask 5 people to solve 6x37 without pencil, paper, or a calculator. Then ask them to slowly go through their mental process of HOW they solved it, and you will probably get 3-5 different trains of thought.
(6x10)3 + (6x5) + (6x1) or (6x3)10 + (6x7) or ....etc. ..It's how YOU think Math. If you understand Math and understand number sense (NUMBER SENSE is what SO many children have problems with - a middle school teacher, myself) you GET Math.
Very interesting and eloquent argument in favour of a critical approach to maths. However, this approach is a threat to the elitism of the current education system. If this was adopted in the way that you mentioned (not only teaching this way, but testing it too) then schools, particularly elite schools, would cease to be exam factories in which those from wealthier backgrounds can be hot-housed to pass these logical tests. I live in the UK and it almost exactly the same here.
Very interesting. I'm a math tutor at my college and, while not a teacher and as a result more difficult to impart (especially at the college level, with so many habits already picked up) is math as a meditation.
Critical thinking as a part of mathematics is essential to mastering mathematics, however I believe much of the problem is primary education's aversion to critical thinking.
I thank you and MJ McDermott for placing forth arguements and opinions on both sides of this debate in the math community (parent and teacher).
In my opinion, you had a better thought out rebuttal to the lady's proposition.
Here are my suggestions: If we wish students to understand math, they first need to have a grasp of English grammar, logic, and rhetoric. Desiphering alogrithms should not be a substitute for the learning of logic. Which, after all, is applicable to all subjects.
I am glad to see that educators recognize that students may be lacking in the critical thinking department. I'm just not sure this new technique is the way to go about it. It seems like a bit of a risk when you're teaching something as fundamental as multiplication and division.
At the end of the day a student's ability to physically interpret mathematics will come down to the talent of the teacher and the student's desire to learn. Just like it always has.
I agree that the most important features are the student's effort and the teacher's gifts. But the program isn't irrelevant. A bad program plus good teacher and good student still won't be nearly as good as good program and good teacher and good student.
May as well come up with a decent program so those that want to learn and those that want to teach will have thoughtful stuff to work with.
Reason why is because I think an intuition teaching method is *already* employed by those teachers that see math through this. I agree that kids don't trust there own thinking. I took traumatic leaps into stochastic processes and data structure/analysis because I employed what works best for me *not* some BS external teaching method.
The way Math was taught in my fathers day, sent man to the Moon and many other things were acomplished with OLD Math. Now that these nerw forms of teaching are in place where does the USA rank in Math on the World Stage? Does anyone have that information? You already know the answer.
@nineteenEcho, I do indeed, and it's not good. But you are missing some important information. The USA dropped in rankings long, long ago. These changes in the math teaching are a RESULT of this decay, not the cause. While you can argue that it isn't working, and the old way is better, what you can't argue (with any reason) that these methods caused the problem.
@jamesblackburnlynch You are absolutely correct. Unfortunately, people DO make that argument, constantly. People who are comfortable with the status quo, whether it works or not, will always be quick to point the finger at any attempts at reform. Then they'll try to reverse any changes to the system and declare themselves reform advocates. It's an intellectually dishonest argument, but it seems depressingly prevalent.
We replaced elemtary arithmetic by machines to get to the moon, (remember slide rules?). Simple arithmetic isn't a competitive advantage. Higher math is where the scientific contribution to economics comes from - try Building the National Ignition Facility, or doing modern genetics, with slide rules.
I teach, among other things, scientific computing. To INVENT algorithms, you cannot only memorize old algorithms. You have to know WHY they work.
I never learned my times tables when I was supposed to. For some reason, it didn't stick, and as I was used to understanding everything EASILY I didn't bother to work to learn them either.
Several years later, after many years of doing multiplication by adding and adding - hey, I had them memorized! I'd also worked out that fractions = dividing on my own, something which lets me do mental math which some people marvel at. (This is learning the HARD way!)
I do not see how there could be any question of if kids should memorize multiplication tables. Multiplication of single digit numbers really should be instantly recalled without any thought. We do the same with addition when we stop adding on our fingers.
Well, as I said in the video, I do think that the kids should memorize single digit multiplication. I don't have a great defense for why, but it's very easy to do and does make some things quicker. My point remains that this is a very small (and trivial) part of mathematics and should not be the focus of the classroom. If it is, it sends the message that mathematics is something to be memorized, not understood. Continue this message throughout, and you have what we do today.
@jamesblackburnlynch The beginning of the video seems to imply that multiplication tables are an example of the bad form of memorization in math. Lots of things really do need to be memorized in math. It isn't memorization that is wrong it is memorization without understanding. I do not think I disagree with what you say in general, I just think that multiplication tables are a poor example. Memorizing the rules of fractions without understanding may be better.
@Tupster, This video is a response to a video whose central question is: "Do you want your child to have memorized their multiplication tables by the time they are in 5th grade?" Using this question as the theme, I am responding to it. In the end, I conclude, yes, we should expect that. But that isn't the important question. Memorization of such things is an insignificant part of what math is and shouldn't be the central question.
@Tupster, As for fractions, that is much more complicated than multiplication. The book I have is for 3rd graders. I want them understanding (and expecting to understand) math before they get to fractions. So, when those rules of fractions come up, they understand they are being taught a trick (common denominators) that is very useful. So, useful indeed, it is worth memorizing, so you don't have to think of how it works every time. But it shouldn't be a mystery either.
I know this is about Math, so don't remind me. To me this is another reason homeschooling is on the rise. When my father, uncle and their friend were in school, they brought their RIFLES to school everyday for ROTC and their SHOTGUNS too so they could walk home and bird hunt. In the 80's everyone at my school (black & white) during hunting season had their guns in their truck back window gun racks. We NEVER even heard of a school shooting, or a racial incident. WE Had corporal punishment.
I looked it up. Corporal punishment is still legal in Louisiana. Has anyone done a study of the states with it legal vs. the ones where it is illegal, in terms of education? That would be interesting.
@jamesblackburnlynch You mean, that would be your next target state. We don't want your help here. We don't have too many problems with out of control kids at our schools. The paddle isn't used much, because the kids know it will be used when needed.
I homeschool my kids now for many reasons.1) This new math 2) No history of the Am Rev War and our Founding Fathers, Brothers & sisters 3)Pushing the Religion of Evolution down my kid's throat 4) Zero tolerence for fighting to the point that a child who defends themselfs are in as much trouble as the Bully 5) Lack of corprol punishment 6) Teachers who have no degree
Well said. From the standpoint of an engineering student finished with Calculus I, II, III and Differential Equations 1, I fully agree. Too often (especially in higher math) theory is often replaced by memorization with no time taken to actually analyze the significance behind what you are learning. For example, why the derivative of x with respect to x is 1, why the integral is x^2/2 + C. Without this analysis, the student just turns into a slow calculator, unable to understand the meaning.
in efforts to make improvements in our society, we have gone to an all or nothing mantra. the eduction system has gone away from rote memory and gone to a theory based learning system like everyday math. A happy middle ground needs to be attained for learning. i meant that there are so many people putting input into the fray that the powers that be swing from exteme to extreme. I did not want to type this much so i cut out a lot.
But EM isn't "theory-based learning." It is application based on learning. So, the idea is to understand the concepts through applications (like the whole part at the end that made MJ so mad). Theoretical learning would be like it was in the 70's with the actual "new math" which was all based on set theory and so on. This stuff isn't theoretical. It's much more applied and hands-on then the old memorization of multiplication tables method.
I stand corrected. Call it application based. I did not intend to quibble over terminology. I have taught EM and did not care for the pase of the course but did like some of the activities and application. I agree with the man who posted this vidoe. Do you like EM?
Aren't I the man who posted this video? Or are we talking about another one? I do like EM, theoretically. I have a couple of videos where I go through the books (which I finally got my hands on). But, as I know, reading a book isn't the same as teaching with it.
change would take place quicker if less people (systems) were involved. (since i know everthing people would be better off just listening to me **this is sarcasm**)
Amen! Thoughtful and well said! Thank you for your articulation of what's needed for a full understanding of mathematics, and the challenges in trying to provide just that.
Because of the piecemeal understanding of what all is needed for children to fully learn math, it takes a host of materials to accomplish this, and this assemblage relies on each teacher's strengths in math, which at elementary is quite spotty. [(Check out a T.Ed dept's elem. math instruction - just laugh (vs. cry.)]
You should look into the Montessori method of teaching math. Every concept is taught following a progression taking students from concrete (hands-on manipulatives) to the abstract. For instance, when studying multiplication, students actually manipulate the materials using repeated addition to solve problems. Eventually, when they are taught algorithms (which, many have discovered on their own), students easily transition because they now have a "short-cut".
Good man! Making kids understand math in the way he describes will be my proffesion one day. learning by understanding is commonplace in our danish schoolsystem, hope you resolve your troubles over there.
The "cluster" method mentioned is brilliant. It's just an the distributive property. It's how I do all my arithmetic. It's never to early to start learning ring properties, IMO.
The problem that is being addressed how been around for a long time. Should we continue to not teach students what they need to know because past generations have been misled?
I do think these books need to have support for parents (and teachers). This must be done on the local level, I believe.
I would say the problem is much worse than it was before - It's the basics they're not getting. Kids can't even count change back to you. They're teaching this stuff in Iowa's schools and our ranking has plummeted on math scores as a result.
First of all, what "basics" do you mean? If you are measuring the kids success by the old rubric, of course they will not be doing as well. These books don't focus on the same material. But that doesn't answer the question of whether they are working or not. To me, the old "basics" weren't good enough.
As for the change thing, neither can adults. That's because many don't understand math. It's not because they can't subtract.
Many other states us EM. Has IA plummeted vs. them?
Okay, I've read the letter. What does it have to do with the change making? I've encountered people who can't make change many times. It's not that they haven't memorized how to subtract, it's that they don't even know they should subtract. That's because they don't understand what subtraction is for. I have students in my class every semester who can't do well in any math class. Yet they can multiply very quickly in their heads. It's simply not the problem.
From the link you mentioned: "Polynomial division is necessary in calculus for factoring. It is used in power series expansion, Laplace Transforms and optimization problems." Is this really a concern for most people? Should we spend a great deal of time on something that will only be of use to the few people who need that? They can pick it up easily when it's necessary.
Not that I'm against learning long division, but that's not much of an argument.
By the way, have you noticed that link was from 2002? Look up Broward County schools now that the curriculum has been running for more than a year. You can see for yourself how they are doing. The No Child Left Behind business requires them to post all their testing in all the grades they do it in. Interesting reading. They do still use EM, I see. But they also offer lots of support on their website for parents.
You note in the video that some teachers do not agree that mathematics is about understanding mathematical ideas. What is a school district to do when 10% or 20% or more teachers hold this view?
I don't really know, but I would celebrate such a district of elementary teachers. I would place the likely number at more like 50-70%. I guess it's not quite that they don't agree...it's that they don't really know what that means. Most of them have just rotely memorized math, from my experience. Oh, and tend to hate it.
I agree about eh difficulty of having elementary (or even high school teachers) who would focus on a deeper understanding of mathematics. Sounds like an impossible dilemma to address. I've always been amazed at the number of middle and high school mathematics teachers who have memorized facts and are unable to appreciate deeper understanding.
I don't have much to say on it, I suppose. But how often do people really need any mathematical procedure at all (other than those that have long been doable by a calculator)? I think that's the whole point here. Most people don't actually ever use the quadratic formula or the law of sines in their whole lives. The point of teaching that material is more about teaching the concepts of mathematics rather than a particular skill set. I think at times that is forgotten.
It's kind of like people arguing about what books to have kids read when they are learning to read. It's not really which books that matter, it's reading. Of course, it'd be nice if the book is engaging. And so with the math.
And relevant, as much as possible. More stats, less algebra!
You have an interesting take on the importance of concepts . . . one that I agree with. However, this is a difficult point for the general public as all math involved was following procedures. Locally, school board members count the number of procedures and concepts (in the district curriculum guides) and expect more focus on the former.
I'm not sure I follow your exact point here. So many school districts have gone with EM, and if they used the algorithm you mention above, I don't think they would have done that. As many many people have complained, that series does not have as one of its strengths "the number of procedures" (unless they count each procedure for multiplication as a distinct one, which would be kind of funny).
But is there really such a simple formula for school boards?
The dilemma is trying to "count" concepts and "count" procedures. Due to the mathematical limitations of many adults, they result to doing word searches. If "concept" shows up more than "procedure" then they conclude curriculum most be poor and unbalanced.
This is done for the entire K-12 curriculum goals. If too many concepts are identified, over procedures, then we need to reduce the number of concepts. After all, didn't the National Math Panel say there needs to be a balance? This is the reasoning of lay people (i.e., school board members). After all, what knowledge do they have of K-12 mathematics?
The difficulty lies in the mathematical knowledge of the BOE and adults in the general populace. Unfortunately, there are many websites that spin the "back to traditional algorithms" web. It's very successful because this is what mathematics is to most adults. Perhaps we need more research on the sorry stated of adult mathematical knowledge to drive change. As it is now, too many adults seems to believe that in the "golden era" of the 60s and 70s that so much more math was learned.
What impact will websites such as wolfram alpha have on the teaching of mathematics? It will do about procedure learned in K-12 and many that are the focus of calculus classes in college. How much time should be spent on procedures when such technology is available? It seems like students could learn much more and better mathematics if we did not spend so much time learning procedures.
I don't think everybody should go to college. It's now where Universities are nothing more than an extension of high school & a business. Ppl just think it's something you do. Too many don't learn how to think & are nothing more than a blank disc & just try to repeat what's been put on it, thinking cannot be taught. Life itself should make one calculate. U.S. society itself with the way it functions from home to social situations is set up to produce idiocy on a mass scale.
I have worked with students in the classroom for the last four year for 3 to 5 hours a week. I can tell you students don't find them intuitive. Since you agree they must ultimately learn an algorithm, why teach a highly inefficient algorithm. When they make a mistake, which happens frequently given the multiple steps, it is too many steps for them to find their error.
Why do you have to make videos teaching the method if it is just so apparent to students?
First of all, you misunderstood what I said. I made videos about Elementary Math: what's in those books and why. I didn't attempt to explain how the different algorithms work. I wouldn't mind doing that actually...books don't explain themselves completely, but I didn't do that.
Secondly, you and I are having a fundamental disagreement about what we want to teach. I want understanding first and preeminently. Mastery of skill second and subordinately.
So, the difference we are having is causing our different points of view. I don't think of the algorithms in terms, simply, of "efficiency." I think of them in terms of understanding. I want students to understand them. The partial products is the easiest one to understand (no hidden meanings in the carrying of digits). The others are used to push students to think about how algorithms are built. It's all about understanding.
Later, the mastery comes. I don't care which algorithm.
The math part of the swedish equivalent of the SAT is done in an interesting way. You are given a problem. Then you are given two additional facts 1 and 2. Your task is to figure out if the problem can be solved using
A. (1) but not (2)
B. (2) but not (1)
C. (1) and (2) together
D. (1) and (2) each by themself
E. Can't be solved using the two facts
That is in my opinion an excellent way to test understanding of math.
In a shop, flour was sold in packages of 2kg, 5kg or 8kg. The difference between the cheapest and most expensive price/kg was $2. Which of the packages had the lowest price/kg?
(1) Buying 2 packages of 8kg cost $4 more than buying 2 packages of 5kg plus 3 packages of 2kg.
(2) The 2kg package had the highest price/kg.
The answer is that you need both (1) and (2) to solve it. So C is the answer.
Everyday Math teaches as many algorithms as classic math, it just teaches highly inefficient algorithms. Whereas classic math maybe takes two steps, the Everyday Math algorithm takes 5, 6 or even more. This does NOT make math more intuitive. It makes math longer, more difficult, more error-prone and more confusing. Everday Math fails to teach math logic or basic math skills. It simply fail students. I recommend you look at the textbooks before you defend them. You might learn something.
You are watching a video that is over two years old. Not only have I actually read those books, I've made videos detailing the materials inside them.
The partial products algorithm, in particular, is in my opinion, everything you claim it is not. It so clear and intuitive that many people come up with it themselves, and don't bother with the "standard algorithm" unless the numbers are very large.
And, in the end, I believe the students must master one of the algorithms.
I'm just kidding about that last statement. But overall I'd much rather have them ask "What does this have to do with math?" instead of "What does this have to do with ANYTHING?" Most textbooks don't do much to alleviate the latter—this is very hard for a textbook to do, quite frankly. Any one that manages to do so gets my kudos--in my book, they have accomplished something very big!
I'm not sure I understand the distinction. Everyday Math is definitely focused on the applications of math. I'm not sure what the other option would mean.
But if they happen to ask this question ("What does this have to do with math?") on a day when I am not particularly prepared, I just tell them to shut up and have fun with it.
That's too bad. Many students will be completely turned off by that answer and conclude that whatever math they are doing that day has no reason at all. For most people (sadly) math is not fun. Even when they understand it, few people actually really enjoy doing math. I'd have to say that most math majors I know don't seem to truly enjoy math.
I sure do. And I've met many who do too. But most, nope.
It is a big victory when a student asks "What does this have to do with MATH?" It tells me right away that they have already accepted that the given scenario does indeed deal with something outside that world of meaningless numbers and manipulations. Confident that the proper content will emerge, I can just smile and ask for patience (hopefully this patience is earned as the course progresses).
Hmm....I'm fairly certain I'm not getting what you are saying here. When do they ask that? When are you talking about something that doesn't appear to be about math? Is this when students are confusing arithmetic with math? That is, if it isn't computation or arithmetic, it's not math?
I'll give you an example. One assignment I gave last year was to fill out an 8-question "Which do you like better?" with two possible choices per question: Coke or Pepsi? Pancakes of Waffles? Wii or Xbox? etc. Initially this assignment appears to have nothing to do with math, prompting some to wonder why they are doing it. At this point, they do not yet know that they are going to translate each of their responses into 1's and 0's and learn about the binary number system.
A project like this is likely to elicit a "What does this have to do with math?" response at the outset. They wonder how their opinions about sports and candy could be related to anything mathematical. When I hear a question like this I know I have already grabbed their attention to some degree. However, when I hear the other question, "What does this have to do with MY LIFE?" this tells me that they are whiny and bored. I most definitely prefer the former.
As a high school teacher I strive to teach my students the skills the state thinks they should know. But even more I want the reasons for those skills to be tightly intertwined in the delivery. I try not to give my students any opportunity to question "What does this have to with anything real?" If a student lacks motivation (as many still will), he or she should be forced into admitting laziness rather than being able to pin it on "irrelevance" of the tasks they are assigned.
Agreed. I find that students are most likely to say "when will I ever use this in my life" when they don't understand the topic. No one asks me that when they get it. That comes up as a defense for something difficult.
In real life math is hardly ever handed to you with pre-constructed problems. A meteorologist I'm sure can attest to this. In real life it is you who defines the problem, you who wrestles with it. Real life math in many cases DOES look like geography on the surface (or science, or humanities, or economics, etc.) It does not make sense to me how someone in a math-related field, who works with maps every day, can see a few maps in a textbook and assert immediately "This is not math."
100% agreement on that. It's as if she didn't actually look at the book there. It's all graphs and percents and fractions and good stuff. And all related to this one topic (the world tour) so that the teacher can keep coming back to the same basic application with many different approaches. There's even some meteorology in there.
The meteorologist, who likely had had an affinity for math all her life, denounced 15-25 pages of the textbook asking "Where's the math?" as she flipped through them quickly. Why? Because they were colorful. They were interesting to look at. They had maps--"that's not math, that's geography!" She did not bother to read or explain the activities associated with those maps, only that the pages were not "chuck full" of written exercises and practice problems.
Yep, agree again. This is what I meant above. Many people think if it's not straight computation it can't be math. They miss the whole point of what math is. And that computation is the final part of a math problem, and is usually the easiest part that a computer could handle, in real life. But the actual person has to get the problem to the point where a computer can handle it. That's the real math.
If math really is everywhere, like we want our students to believe, then a textbook that aims to provide a rich, meaningful context for students to practice real math must certainly be a step in the right direction. I can't say I vouch for throwing the traditional algorithms out the window, but I do disagree with the idea of discrediting a resource on the basis that it looks so different.
And the book doesn't throw the standard algorithm out. It eventually concludes that it is the one to use, for most people. But it works up to it, and demands that students understand it. One of my favorite parts of the book is that when it brings up the lattice method, it doesn't explain it to them. It shows an example and asks them to figure it out. Any student who can definitely understands the standard algorithm. Very challenging question.
Is math really its own separate world of thought, void of apparent meaning, that some get and some just don't? Of course not! I can't imagine any educator believing this--but the problem is that many students do--the majority most likely.
I agree that imparting a love of math to many is difficult; and I believe it is downright impossible with the approach used in most classrooms: "Math is what it is--working with numbers on paper. You may love it or hate it, but either way you need to be good at it." Hard to get people to care about math with this mindset. Of all the disciplines, it is perhaps the least challenged by adults in terms of its necessity in education--but also the least understood by students as to why.
Hello. I've been working as a math tutor (elementary algebra through calculus 1) at my college for several years, so I also get a lot of first-hand experience in how unprepared for college-level math some students can be.
I love math because I "get" math -- everything in the universe boils down to math. However I find it hard to tutor some students because I find it difficult to impart that intrinsic understanding of math. Maybe it's because of the way I was taught in school.
I was taught to parrot multiplication tables and apply them to multiplication and division like anyone else my age and older was. I personally see it as kind of a shame when a student has to use a calculator for while I consider a simple multiplication such as 6x4. Maybe I expect the student to have memorized it like I did, or maybe I see the calculator as a time-waster rather than a time-saver for simple arithmetic, and I always teach my students that speed is important second to accuracy.
As far as formulas go, a sad truth about math is that there *are* a lot of formulas to memorize (though I don't have to tell you that) so you can never get away from memorization. The Pythagorean Theorem for example -- I can understand right triangles and how one angle is 90 degrees and the other two are complementary, but if I don't know a^2 + b^2 = c^2 and that c is the hypotenuse, I won't be able to tell you the missing side of that triangle.
The Pythagorean Theorem gets such a big name because it is such a significant result. The proof is not just common sense. I agree it does have to be memorized in the end. But I don't agree that there is a lot to memorize. Well, not compared with what people think.
For example, I got into a conversation here on youtube with a student. She believed she was being asked to memorize what 21*56 was! It's hard for students to distinguish what is to be memorized and what is to be figured out.
I agree with you 100% that students need to be imparted this distinguishment, but it is a bit frustrating to see students trying to memorize things that cannot be easily explained. For example, I can't prove to you that pi*r^2 is the area of a circle without deriving it using limits, but the students that would understand that derivation would not need to be proven that pi*r^2 is the area of a circle.
So regardless of your level of understanding in math, there is a lot of "this is how it is."
Yes, pir^2 is another one that is difficult to explain. Again, this example is from geometry.
But these are exceptions. I'm teaching College Algebra right now and there is virtually nothing I need to make them memorize.
The quadratic formula...I derived this for them using completing the square and so on...but I told them that this is the rare time I think it's better just to memorize. The derviation is too difficult to redo all the time. And it's not very revealing. But what else?
Lately I've been working with summer students on linear equations.
The standard and general forms are probably the most confusing because it is hard to parse anything meaningful from them without plugging in an x to get a y and plot a few points.
Whereas slope-intercept form you can plot several points right away just by knowing the y-intercept and slope. Point-slope form is also meaningful because it still gives you the slope and a starting point.
I agree completely. I always tell my students that the standard form is basically useless. It hides the slope and the y-interecept or any other point. The slope-intercept form is the one that makes the most intuitive sense (if you make sure they first understand that lines are models that describe a constant growth/decline by addition/subtraction). I tell them that I don't bother memorizing point-slope but I see the advantage to it. But standard form? Pass. (Only for vertical lines.)
After watching more of your videos and talking to you some I think I agree with you -- math education needs to change on a fundamental level. If mathematical competency were to improve, then so would competency in every other subject that directly involves math: physics, chemistry, astronomy, etc.
However it is hard to change an education system. The most I hope to do is impart a love or at least an understanding of math to my students.
Imparting an understanding is all we are after. Imparting a love of math would be nice, but it's not at all realistic to think you can do that with many. We can't really all love the same subjects. But the idea that students could imagine why someone would really enjoy math, even if they don't, would be nice. That basically can be passed on just by loving it and showing that love each day. And making sure that it makes sense.
I know I really shouldn't post this because it's so stupid, but it is the truly correct answer to the person-cave question you asked...
They can't go all the way through the cave because caves have an end! It'd be a tunnel if they could go all the way through! That means it doesn't even matter the average height of the tunnel or how tall the person is! That's the real answer to your hypothetical question!
I can't have the t-shirt you're wearing?! Well... What about one of your shoes? I just need one. Preferably your left, unless it's a slipper.
But I do agree with what you're saying about those books. They're trying to teach WHY an answer is what it is, and not just what the answer is. But the issue is that they don't bother to teach the easier way of solving equations once the children get the reasoning behind solving the questions. Instead, they just say to use a calculator.
If that really is so, that's not okay with me. As I said in my videos, I think the focus on understanding is essential, but I also think, that once the understanding is solid, then practice is essential too. Maybe calculators obviate the need for the standard algorithm (I don't think so, but let's just say they do), but the importance of practice is also a message that must be established early.
Talent is great. But talent and hard work, that's what it really takes.
I am also disturbed that most elementary school math programs put emphasis on memorizing specific techniques rather than on finding solutions through the use of solid reasoning. While I do think teachers should still teach the traditional multiplication and division algorithms, I have a problem with them in that it is nearly impossible to explain to an elementary school student WHY the traditional algorithms work. The reasons are more complicated than one might think
Even the average high school senior would have trouble understanding the derivation of long division. This is another reason teachers should teach some of the other methods mentioned in the video. They actually make sense!
I think the standard algorithm for multiplication isn't too complicated, particularly if one starts by looking at the partial-products method. The division algorithm is perhaps a bit much.
Like many of these respondents, I wanted to thank you immensely for saying this. "An Inconvenient Truth" struck me very much as an argument for stagnation; primarily the message seemed to be one in which you teach what's familiar, even if it's known that what's being taught fails a great many people. I believe that rote mathematics can be useful for quick answers, but agree with you that rote does not teach the concepts behind mathematics, something which is sorely lacking in math education.
Please communicate the importance of reasoning with others. In Seattle, WA and other places, parents have banded together to disparage reasoning and thinking. They emphasize the use of multiple examples that students should mimic. They say that students should learn "concepts," but never describe what these are. Instead, the encourage moving back to traditional texts so that the "tried and true" will remain. I guess that means our jobs are safe. Few will learn math with deep understanding.
I've tried to do that. My hope with this video was to have a dialogue with people about this issue. I have for a couple of years now. I doubt I've changed minds (nor has anyone really changed mine) but I've had some interesting discussions and I understand better why they believe what they do.
I did watch a Saxon video and they certainly do sound like they care about the concepts. The problem is their system doesn't really make the students learn it. They can succeed with pure memory.
Just wanted to say that I think you have it exactly right. She isn't comprehending the larger picture either. The different methods are good for helping the student to 'understand' the math rather than just memorizing but the system on a whole must be one wy or the other. As you point out, if they are taught one way and then tested on another, everyone becomes confused. Glad to see an intelligent reply to her video.
Your video response was useful. I believe that students should be able to understand how math contributes to daily lives and how to utilize it. My friend told me how our teacher never tells us why we do some techniques. She only tells us that it is needed for our test. We agreed together that our teacher is not a superior math teacher. She only gives shortcuts to problems, but never the reasons why we do it. Maybe its because she only teaches high school, but that should not be an excuse.
I think what heoTheo meant was that even though carrying numbers in mathematical processes, namely multiplication and addition, it would be better to develop a way to avoid this type of algorithm especially when dealing with large numbers using only pencil and paper. unless heoTheo was just goofing around, in which case forget what i thought he meant hehehe
Well, if that's the case, then the cluster algorithm is close to what heoTheo wants. "Carrying" is hiding that you are adding a tens digit (or hundreds, etc.) and the cluster algorithm refuse to hide it.
I just wanted to say thanks for making these response videos. You say exactly what I felt when I watched the 'inconvenient truth' video.
I'm actually a third year physics major, and I've thought on and off about going into education, but I've always been conflicted with the issue of perhaps being restrained into teaching to the test and such.
Anyways, thanks again for taking the time to make these responses.
I found this kind of problem with physics, rather than basic math. I could grasp math easily, because I could work out how everything related. The problem I hit with physics is that suddenly I was spoon fed tons of equations, and while sure I understood many parts, it mostly seemed completely arbitrary and something that merely needed memorized. I found that it became far easier to understand when I understand the history of the discovery of the equations, but it was rare to get that.
This is a different topic, but I think I understand what happens in physics. I took a physics class a few years ago to see what's going on there.
In physics, the mathematics of the students is always several years behind the physics. So, the physics teacher has to fake the math they use. For students with an intuitive grasp of physics, they can just ignore this, but for the more mathematical mind, it can drive the student crazy and prevent understanding.
A lot of my teachers actually do show the derivations of the formulas, and then explain how we get there, but the problem is that nobody pays attention until the part where they get the final formula, and then they just go and memorize it.
How would you suggest we solve a problem like that?
Yes, I call this the "blah blah blah answer" interpretation by students of a lecture. I've checked the notes of lots of students, and this is the norm. They don't bother writing any of the concepts/big ideas down. They think just the formulas and an example will do.
How do deal with this? Not easy, but the first step is to make sure that doesn't work. That is, test problems must be such that memorizing formulas won't get you there. They must challenge understanding.
I think you're leaving out one factor: the actual students. Some students WON'T 'step back' and try to understand it. They just want to memorize the formulas and pass the test. It's very hard to make someone understand something when they simply don't want to, I've seen it in a lot of my classmates (I'm a grade 11 student in Canada).
I never memorize formulas for tests, I can't remember, I'll just derive it.
It's true that some just want to memorize. Or, really, many many do. And if it works, what's to stop them? It's important to show them that it doesn't work. I think the teacher actually has to have an honest conversation where students get to argue that shouldn't have to do anything more than memorize. And then the teacher should make their case that that isn't enough.
What's more, some students who want to learn more, don't really know how to. They want to...but don't know how.
It depends at what age you get them I think. I think all children are naturally curious until it gets beaten out of them in the current teaching methods..
You say that the idea behind "these books" is to teach the idea behind the algorythm. But I don'tsee how these other methods teach the desired idea at all, either.
Her point on the original video is that the common algorithm is simpler and better. Neither teaches much of the idea.
I guess I am thinking you might be a little naive in assuming that "these books" are trying to teach the thinking. I see no sign of that.
Have you seen my other videos where I discuss the books themselves (which I finally got my hands on)? There I detail specific ways the books get at the concepts.
For this particular one, the cluster algorithm, in my opinion, is the idea of the standard algorithm, but without the "mysterious" carry digit. It clarifies the whole procedure.
The lattice method is used in these books to push even further the understanding by coming up with a defense of a strange algorithm.
I agree that word problems (or, to put it more simply, real-life problems) should be the goal. But, as you probably know, students hate word problems, usually. That's because word problems require thought. And we seem to keep telling our students that math should be free of thought.
Is that a satisfying answer to the question? I think most of my students would find that answer very off-putting. To me it sounds basically like "you won't ever need this."
I guess my question would be this...you talk a lot and make very valid points about those who just "pick things up" vs. those who don't. And that the program needs to work so that the ones who can't just pick it up automatically aren't having a disservice done to them. But is it doing that. I've recently helped a kid by teaching him a way to do fractions that he got. But his teacher told him he was doing it the wrong way (not her way). Even though it works all the time. How is that helping?
Clearly, that's not helping. But it's also against the philosophy of the book. Multiple approaches are more than welcomed, they are required. Now, it's possible the teacher was saying that the student needed to be able to do it the way you showed and the other way too. But if she told him her way was the only acceptable way, then clearly she doesn't believe in these books.
But she is using the TERC method and she clearly believes in it. This child got the highest score on his latest test, after I taught him to understand in 10 minutes what his teacher couldn't teach in 2 weeks. And clearly, from the scores, the other kids didn't get it either. Perhaps this is just a bad math teacher, but I am not a math teacher at all, and yet I have taught this child fractions where his teacher failed. And he understands why with my method. And he gets the right answer.
Again, my question would be, if the majority learn "why" just by learning the formulas, and from my experience in college, most did, why change the method so that the minority might learn why, and instead no one learns at all. I get why it is important to understand the why. But I don't agree with the argument that the old methods don't work to teach this for the majority. And again, I am not a math teacher, so obviously my view is a bit off.
why would the teachers not think the books are a good idea? memorising formulas will only get you so far you need to learn problem solving skills think logically but also if everything is in a context would it not make students think what is the use of this when maths isnt just applied to use maths is also theoretical
MathMikie 2 minutes ago
Also, nice example of dividing by zero in the first part. The way I always thought of it was that you can't take something and divide it among zero people. You can divide it for one person by leaving it as a whole, but not among zero people.
scoobertYTP 1 month ago
@scoobertYTP Okay, but then think about this...why is 0/0 undefined? Certainly you can divide 0 things up among 0 people, right?
jamesblackburnlynch 1 month ago
@jamesblackburnlynch Hmm, never really thought about that one before. I was going to say that you can't divide anything among zero people in the first place because, if you were going to divide anything (even if its amount is 0) among anyone, there would have to be at least one person, but I understand what you're saying there (like if you have nothing to divide). And I guess it doesn't really work as well as what you said because you can divide by a number smaller than 1.
scoobertYTP 1 month ago
@scoobertYTP If you think about it the way I described for 1/0...that is, 0 times what equals 1, then for 0/0 the question becomes 0 times what equals 0? And the answer this time is...every number there is. Anything times 0 will give you 0. (Which matches the idea that if you 0 things you can divide them up among 0 people anyway you want...since there are 0 things. Sort of. Some of the intuitive nature of the 1/0 as breaking up into groups is lost.) Too many answers to define!
jamesblackburnlynch 1 month ago
@jamesblackburnlynch Interesting thing you're pointing out there. If you think about it that way, anything times 0 does give you 0. Yet you can't divide by 0. I sometimes think about it was as if you actually had to give something (or nothing) to someone by dividing. If you tried to give it to zero people (including yourself, which would be 1) then you couldn't do it. And it sort of works for numbers between 0 and 1 because you're multiplying instead. But what about negative numbers?
scoobertYTP 1 month ago
@scoobertYTP I can't pretend to understand what you are saying there. Division really is what I'm saying. I'm not coming up with anything clever. 6/2 means "2 times what =6?" So 1/0 means "0 times what =1?" As we've said, that's impossible, so 1/0 is undefined. And 0/0 means "0 times what =0?" That's everything. Since mathematicians want a unique answer to every question, and no answer is any better than any other here, we choose not to define 0/0. That's all.
jamesblackburnlynch 1 month ago
@jamesblackburnlynch Eh, the thing I'm saying is extremely complicated and is very hard to explain, especially since it doesn't really work for explaining negative numbers. For some reason it always just worked for me. But I agree completely with what you're saying division is. And now I understand completely what you mean with why 0/0 is undefined. There are infinitely many solutions to 0 times what is zero, which means there are too many solutions to define one.
scoobertYTP 1 month ago
@scoobertYTP Well, that's exactly what I mean, so yep, what you are saying is probably just too hard for me to understand in 400 whatever characters. Now...why is 2^0=1? Enjoy...
jamesblackburnlynch 1 month ago
I agree about your ideas on what math should be. Up until about a year before I started college, I was one of those people who thought that math was just a bunch of formulas that really didn't relate too much to life. After that I realized that, if you don't just do it by memorizing formulas, it causes you to think in a certain way and helps you apply concepts involving logic that could be valuable in the future. I'm better at math now, but I still wish I would've realized this sooner.
scoobertYTP 1 month ago
You just confirmed her argument with your closing statement. Everyday Math is concerned with understanding math ideas WITHOUT spending time to master them. So in essence, there is a problem with those books.
gothamvengeance 2 months ago
@gothamvengeance Actually, her argument has more to it than you think. She is arguing about the focus of math education. She wants it to be on mastering efficient skills. It's true that she wants them to have more time practicing, but she also wants students to be shown what is the "best" way without wasting time on less efficient methods. That is, she sees math as a collection of skills to be mastered. I don't. But I do agree, there's nothing wrong with mastery of skills.
jamesblackburnlynch 2 months ago
@jamesblackburnlynch I appreciate your response. From my understanding, the Singapore method (which is the subject of the books she recommends at the end of her video) has a balanced focus on understanding the ideas *and* mastering the skills.
I am a parent of a 5th grade daughter who is struggling mightily with the Everyday Math program. I agree that basic critical thinking should be a goal of what is being taught, but not without the mastery of doing so.
gothamvengeance 2 months ago
@gothamvengeance I might have to watch her again, but I believe the books she recommends are the Saxon books, which I do not like at all. They are all about skill practice.
Singapore, on the other hand, I'm a big fan of. It does both the conceptual and skill sides that I want. Over the years, I've come to the conclusion that EM is just too challenging. I think one can supplement the skill mastery, no problem, but there is too much in those books.
jamesblackburnlynch 2 months ago
What school do you teach at???????
SpaWnoFKollias 4 months ago
dude you need to be my math teacher!!!Most of my teachers dont even like to teach math and they get frustrated when helping people.
SpaWnoFKollias 4 months ago
@SpaWnoFKollias You aren't talking about college, right? Most college professors in math love math. Of course, they don't necessarily love math.
My school is Berea College. Fine school.
jamesblackburnlynch 4 months ago
I find the traditional methods simpler to understand and to explain to young children. If properly taught they make sense. There is no more intuition or logic to any of the newer methods I’ve seen. The reasons certain people have issues with math are many and complex. I don’t think properly teaching and explaining the traditional algorithms is one of them.
Eirana9 5 months ago in playlist More videos from jamesblackburnlynch
I have advanced degrees in Math and Statistics and have taught at the college level. I do not think anyone is saying you should just teach algorithms in a vacuum with no explanation as to why they work. If explained properly the traditional methods have just as much, if not more, logic than those of the other methods. I have seen this taught properly to my 4th grade son using diagrams and word problems to show their relevance.
Eirana9 5 months ago in playlist More videos from jamesblackburnlynch
A+ I teach Everyday math at my school. Many teachers struggle to teach it because they don’t truly understand math themselves, which of course leads parents to believe there is a major problem with the curriculum themselves. They try to teach it as rote memory (like “old” math). I’m a math person and developed my own strategies as a child. Non-math people never create these strategies and end up hating math. EDM tries to teach all students what math people have been doing for years.
dclausing 7 months ago
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The problem I've seen with the new math texts is that they often preclude parents (who learned traditional math) from being able to step in and help when their child is having problems. What happens is the parent is confronted with a jumble of terms and methods that they aren't familiar with. Then the parent gets angry because they can't help their child.
This problem will presumably disappear in a generation. But I think it could have been avoided if more thought had been given to this.
kenj0418 7 months ago
I can't stop hitting the like button!!!
MetalSpiral 8 months ago
YES
When I was learning math, the key to my success was just that. Context, algorithm, theory, and practice. Solving problems in different ways was also important. Versatility.
MetalSpiral 8 months ago
I really appreciate this video response. I am a college student with a minor in physics, and as I was listening to "An Inconvenient Truth," I started to feel more and more like the claims in those videos were unjustified. I'm glad to see someone with a real sense of what mathematics is and what mathematics can achieve speaking to the issue.
ez910503 9 months ago
Thanks for your response, by the way what the lady called "standard" it is here, I learn a similar but different way to multiply, that she is more proficient in one way or another that does not mean it is wrong. It is as you say, much better than me, another way to see it to understand better of what is behind the formula
ls604 9 months ago
I find the idea that the weather forecaster presents - which is there is a vessel (student) into which a method is poured - frightening in its ideology - that isn't how education works. The students are not empty vessels and the teacher is not pouring some pure knowledge into the child's head.
johncrwarner 11 months ago
Thank you so much. I am also a college math teacher (after 7 yrs as a h.s. math teacher). This is my first experience with YouTube and I searched on Math first. My daughters successfully learned math with Everyday Math and both completed AP Calculus with 5's. As you said here, people have been saying the same things about math proficiency and education in general for decades (actually, centuries). It's interesting that the original video was not done by a math teacher but a meterologist.
MathLadyDLogan 1 year ago
What a passionate and enlightened dialogue. If only we could have these kinds of real conversations about all of our problems in the education system (math, the lack of civics, etc...) and the problems in our country generally, we would be a lot better off!
middleCmusic 1 year ago
@middleCmusic Yeah, it's good when it's going well. I have to admit I delete some comments because they appear to be just trolling to me.
jamesblackburnlynch 1 year ago
Simple Example:
Ask 5 people to solve 6x37 without pencil, paper, or a calculator. Then ask them to slowly go through their mental process of HOW they solved it, and you will probably get 3-5 different trains of thought.
(6x10)3 + (6x5) + (6x1) or (6x3)10 + (6x7) or ....etc. ..It's how YOU think Math. If you understand Math and understand number sense (NUMBER SENSE is what SO many children have problems with - a middle school teacher, myself) you GET Math.
GREAT video debate
McKennahLane 1 year ago
Very interesting and eloquent argument in favour of a critical approach to maths. However, this approach is a threat to the elitism of the current education system. If this was adopted in the way that you mentioned (not only teaching this way, but testing it too) then schools, particularly elite schools, would cease to be exam factories in which those from wealthier backgrounds can be hot-housed to pass these logical tests. I live in the UK and it almost exactly the same here.
dolgorukas 1 year ago
Very interesting. I'm a math tutor at my college and, while not a teacher and as a result more difficult to impart (especially at the college level, with so many habits already picked up) is math as a meditation.
Critical thinking as a part of mathematics is essential to mastering mathematics, however I believe much of the problem is primary education's aversion to critical thinking.
SuperNegativeCreep 1 year ago
I thank you and MJ McDermott for placing forth arguements and opinions on both sides of this debate in the math community (parent and teacher).
In my opinion, you had a better thought out rebuttal to the lady's proposition.
Here are my suggestions: If we wish students to understand math, they first need to have a grasp of English grammar, logic, and rhetoric. Desiphering alogrithms should not be a substitute for the learning of logic. Which, after all, is applicable to all subjects.
MathewStory 1 year ago
@MathewStory How about both at the same time?
diomedes39 1 year ago
I am glad to see that educators recognize that students may be lacking in the critical thinking department. I'm just not sure this new technique is the way to go about it. It seems like a bit of a risk when you're teaching something as fundamental as multiplication and division.
At the end of the day a student's ability to physically interpret mathematics will come down to the talent of the teacher and the student's desire to learn. Just like it always has.
rossicorona 1 year ago
I agree that the most important features are the student's effort and the teacher's gifts. But the program isn't irrelevant. A bad program plus good teacher and good student still won't be nearly as good as good program and good teacher and good student.
May as well come up with a decent program so those that want to learn and those that want to teach will have thoughtful stuff to work with.
jamesblackburnlynch 1 year ago
Just learning only to use fractions is really helpfull in understanding math.
thehandsomedevils 2 years ago
This guy is a real schmuck...
What he is saying isn't too bad though.
scout6686 2 years ago
why is he a schmuck?
thehandsomedevils 2 years ago
Reason why is because I think an intuition teaching method is *already* employed by those teachers that see math through this. I agree that kids don't trust there own thinking. I took traumatic leaps into stochastic processes and data structure/analysis because I employed what works best for me *not* some BS external teaching method.
scout6686 1 year ago
The way Math was taught in my fathers day, sent man to the Moon and many other things were acomplished with OLD Math. Now that these nerw forms of teaching are in place where does the USA rank in Math on the World Stage? Does anyone have that information? You already know the answer.
nineteenEcho 2 years ago
@nineteenEcho, I do indeed, and it's not good. But you are missing some important information. The USA dropped in rankings long, long ago. These changes in the math teaching are a RESULT of this decay, not the cause. While you can argue that it isn't working, and the old way is better, what you can't argue (with any reason) that these methods caused the problem.
jamesblackburnlynch 2 years ago
@jamesblackburnlynch You are absolutely correct. Unfortunately, people DO make that argument, constantly. People who are comfortable with the status quo, whether it works or not, will always be quick to point the finger at any attempts at reform. Then they'll try to reverse any changes to the system and declare themselves reform advocates. It's an intellectually dishonest argument, but it seems depressingly prevalent.
TransparentChocolate 1 year ago
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zenpharaohs 1 year ago
We replaced elemtary arithmetic by machines to get to the moon, (remember slide rules?). Simple arithmetic isn't a competitive advantage. Higher math is where the scientific contribution to economics comes from - try Building the National Ignition Facility, or doing modern genetics, with slide rules.
I teach, among other things, scientific computing. To INVENT algorithms, you cannot only memorize old algorithms. You have to know WHY they work.
zenpharaohs 1 year ago
I never learned my times tables when I was supposed to. For some reason, it didn't stick, and as I was used to understanding everything EASILY I didn't bother to work to learn them either.
Several years later, after many years of doing multiplication by adding and adding - hey, I had them memorized! I'd also worked out that fractions = dividing on my own, something which lets me do mental math which some people marvel at. (This is learning the HARD way!)
Conuly 2 years ago
I do not see how there could be any question of if kids should memorize multiplication tables. Multiplication of single digit numbers really should be instantly recalled without any thought. We do the same with addition when we stop adding on our fingers.
Tupster 2 years ago
Well, as I said in the video, I do think that the kids should memorize single digit multiplication. I don't have a great defense for why, but it's very easy to do and does make some things quicker. My point remains that this is a very small (and trivial) part of mathematics and should not be the focus of the classroom. If it is, it sends the message that mathematics is something to be memorized, not understood. Continue this message throughout, and you have what we do today.
jamesblackburnlynch 2 years ago
@jamesblackburnlynch The beginning of the video seems to imply that multiplication tables are an example of the bad form of memorization in math. Lots of things really do need to be memorized in math. It isn't memorization that is wrong it is memorization without understanding. I do not think I disagree with what you say in general, I just think that multiplication tables are a poor example. Memorizing the rules of fractions without understanding may be better.
Tupster 2 years ago
@Tupster, This video is a response to a video whose central question is: "Do you want your child to have memorized their multiplication tables by the time they are in 5th grade?" Using this question as the theme, I am responding to it. In the end, I conclude, yes, we should expect that. But that isn't the important question. Memorization of such things is an insignificant part of what math is and shouldn't be the central question.
jamesblackburnlynch 2 years ago
@Tupster, As for fractions, that is much more complicated than multiplication. The book I have is for 3rd graders. I want them understanding (and expecting to understand) math before they get to fractions. So, when those rules of fractions come up, they understand they are being taught a trick (common denominators) that is very useful. So, useful indeed, it is worth memorizing, so you don't have to think of how it works every time. But it shouldn't be a mystery either.
jamesblackburnlynch 2 years ago
I know this is about Math, so don't remind me. To me this is another reason homeschooling is on the rise. When my father, uncle and their friend were in school, they brought their RIFLES to school everyday for ROTC and their SHOTGUNS too so they could walk home and bird hunt. In the 80's everyone at my school (black & white) during hunting season had their guns in their truck back window gun racks. We NEVER even heard of a school shooting, or a racial incident. WE Had corporal punishment.
nineteenEcho 2 years ago
@nineteenEcho, you had corporal punishment in the 80's?? I don't think that was legal...was it? It wasn't where I grew up and that was the 80's.
jamesblackburnlynch 2 years ago
Comment removed
nineteenEcho 2 years ago
I looked it up. Corporal punishment is still legal in Louisiana. Has anyone done a study of the states with it legal vs. the ones where it is illegal, in terms of education? That would be interesting.
jamesblackburnlynch 2 years ago
Well, Louisiana in particular is either THE worst performing state, as far as education goes, or one of the bottom five.
But there's a lot that goes into that, it's not just about how they discipline their students.
Conuly 2 years ago
@jamesblackburnlynch You mean, that would be your next target state. We don't want your help here. We don't have too many problems with out of control kids at our schools. The paddle isn't used much, because the kids know it will be used when needed.
nineteenEcho 2 years ago
@nineteenEcho, I'm not in charge. I don't have any target states. I'm just an interested person.
So...how is your state doing nationally? Is the system working?
jamesblackburnlynch 2 years ago
I homeschool my kids now for many reasons.1) This new math 2) No history of the Am Rev War and our Founding Fathers, Brothers & sisters 3)Pushing the Religion of Evolution down my kid's throat 4) Zero tolerence for fighting to the point that a child who defends themselfs are in as much trouble as the Bully 5) Lack of corprol punishment 6) Teachers who have no degree
nineteenEcho 2 years ago
Well said, thanks for posting your views!
AlexTube2006 2 years ago
Well said. From the standpoint of an engineering student finished with Calculus I, II, III and Differential Equations 1, I fully agree. Too often (especially in higher math) theory is often replaced by memorization with no time taken to actually analyze the significance behind what you are learning. For example, why the derivative of x with respect to x is 1, why the integral is x^2/2 + C. Without this analysis, the student just turns into a slow calculator, unable to understand the meaning.
Griever0510 2 years ago
amen. too many opions out there. the world would be a better place if everyone just did what i say.
kurtzb1970 2 years ago
????
jamesblackburnlynch 2 years ago
in efforts to make improvements in our society, we have gone to an all or nothing mantra. the eduction system has gone away from rote memory and gone to a theory based learning system like everyday math. A happy middle ground needs to be attained for learning. i meant that there are so many people putting input into the fray that the powers that be swing from exteme to extreme. I did not want to type this much so i cut out a lot.
kurtzb1970 2 years ago
But EM isn't "theory-based learning." It is application based on learning. So, the idea is to understand the concepts through applications (like the whole part at the end that made MJ so mad). Theoretical learning would be like it was in the 70's with the actual "new math" which was all based on set theory and so on. This stuff isn't theoretical. It's much more applied and hands-on then the old memorization of multiplication tables method.
jamesblackburnlynch 2 years ago
I stand corrected. Call it application based. I did not intend to quibble over terminology. I have taught EM and did not care for the pase of the course but did like some of the activities and application. I agree with the man who posted this vidoe. Do you like EM?
kurtzb1970 2 years ago
Aren't I the man who posted this video? Or are we talking about another one? I do like EM, theoretically. I have a couple of videos where I go through the books (which I finally got my hands on). But, as I know, reading a book isn't the same as teaching with it.
jamesblackburnlynch 2 years ago
Re: the world would be... part.
change would take place quicker if less people (systems) were involved. (since i know everthing people would be better off just listening to me **this is sarcasm**)
kurtzb1970 2 years ago
Amen! Thoughtful and well said! Thank you for your articulation of what's needed for a full understanding of mathematics, and the challenges in trying to provide just that.
Because of the piecemeal understanding of what all is needed for children to fully learn math, it takes a host of materials to accomplish this, and this assemblage relies on each teacher's strengths in math, which at elementary is quite spotty. [(Check out a T.Ed dept's elem. math instruction - just laugh (vs. cry.)]
pamtube1 2 years ago
You should look into the Montessori method of teaching math. Every concept is taught following a progression taking students from concrete (hands-on manipulatives) to the abstract. For instance, when studying multiplication, students actually manipulate the materials using repeated addition to solve problems. Eventually, when they are taught algorithms (which, many have discovered on their own), students easily transition because they now have a "short-cut".
superpinkybear 2 years ago
I'm familiar with Montessori as a type of education system. Is there a particular book that you are referring to that I could see?
It sounds like a similar approach to what is recommended by those who wrote these books.
jamesblackburnlynch 2 years ago
Good man! Making kids understand math in the way he describes will be my proffesion one day. learning by understanding is commonplace in our danish schoolsystem, hope you resolve your troubles over there.
madshebs 2 years ago
The "cluster" method mentioned is brilliant. It's just an the distributive property. It's how I do all my arithmetic. It's never to early to start learning ring properties, IMO.
aaronasterling 2 years ago
Well, the new math sure keeps parents from helping. How helpful is that?
anonymous915 2 years ago
The problem that is being addressed how been around for a long time. Should we continue to not teach students what they need to know because past generations have been misled?
I do think these books need to have support for parents (and teachers). This must be done on the local level, I believe.
jamesblackburnlynch 2 years ago
I would say the problem is much worse than it was before - It's the basics they're not getting. Kids can't even count change back to you. They're teaching this stuff in Iowa's schools and our ranking has plummeted on math scores as a result.
anonymous915 2 years ago
First of all, what "basics" do you mean? If you are measuring the kids success by the old rubric, of course they will not be doing as well. These books don't focus on the same material. But that doesn't answer the question of whether they are working or not. To me, the old "basics" weren't good enough.
As for the change thing, neither can adults. That's because many don't understand math. It's not because they can't subtract.
Many other states us EM. Has IA plummeted vs. them?
jamesblackburnlynch 2 years ago
Yes it is because they can't subtract.
Do a web search for a letter "From Karen S. Jones-Budd February 12, 2002"
"Bad experience with "Everyday Mathematics" in Broward County, FL"
anonymous915 2 years ago
Okay, I've read the letter. What does it have to do with the change making? I've encountered people who can't make change many times. It's not that they haven't memorized how to subtract, it's that they don't even know they should subtract. That's because they don't understand what subtraction is for. I have students in my class every semester who can't do well in any math class. Yet they can multiply very quickly in their heads. It's simply not the problem.
jamesblackburnlynch 2 years ago
From the link you mentioned: "Polynomial division is necessary in calculus for factoring. It is used in power series expansion, Laplace Transforms and optimization problems." Is this really a concern for most people? Should we spend a great deal of time on something that will only be of use to the few people who need that? They can pick it up easily when it's necessary.
Not that I'm against learning long division, but that's not much of an argument.
jamesblackburnlynch 2 years ago
By the way, have you noticed that link was from 2002? Look up Broward County schools now that the curriculum has been running for more than a year. You can see for yourself how they are doing. The No Child Left Behind business requires them to post all their testing in all the grades they do it in. Interesting reading. They do still use EM, I see. But they also offer lots of support on their website for parents.
jamesblackburnlynch 2 years ago
You note in the video that some teachers do not agree that mathematics is about understanding mathematical ideas. What is a school district to do when 10% or 20% or more teachers hold this view?
sleeper2345 2 years ago
I don't really know, but I would celebrate such a district of elementary teachers. I would place the likely number at more like 50-70%. I guess it's not quite that they don't agree...it's that they don't really know what that means. Most of them have just rotely memorized math, from my experience. Oh, and tend to hate it.
jamesblackburnlynch 2 years ago
I agree about eh difficulty of having elementary (or even high school teachers) who would focus on a deeper understanding of mathematics. Sounds like an impossible dilemma to address. I've always been amazed at the number of middle and high school mathematics teachers who have memorized facts and are unable to appreciate deeper understanding.
*
Any comments on my wolfram alpha question below?
sleeper2345 2 years ago
I don't have much to say on it, I suppose. But how often do people really need any mathematical procedure at all (other than those that have long been doable by a calculator)? I think that's the whole point here. Most people don't actually ever use the quadratic formula or the law of sines in their whole lives. The point of teaching that material is more about teaching the concepts of mathematics rather than a particular skill set. I think at times that is forgotten.
jamesblackburnlynch 2 years ago
It's kind of like people arguing about what books to have kids read when they are learning to read. It's not really which books that matter, it's reading. Of course, it'd be nice if the book is engaging. And so with the math.
And relevant, as much as possible. More stats, less algebra!
jamesblackburnlynch 2 years ago
You have an interesting take on the importance of concepts . . . one that I agree with. However, this is a difficult point for the general public as all math involved was following procedures. Locally, school board members count the number of procedures and concepts (in the district curriculum guides) and expect more focus on the former.
sleeper2345 2 years ago
I'm not sure I follow your exact point here. So many school districts have gone with EM, and if they used the algorithm you mention above, I don't think they would have done that. As many many people have complained, that series does not have as one of its strengths "the number of procedures" (unless they count each procedure for multiplication as a distinct one, which would be kind of funny).
But is there really such a simple formula for school boards?
jamesblackburnlynch 2 years ago
The dilemma is trying to "count" concepts and "count" procedures. Due to the mathematical limitations of many adults, they result to doing word searches. If "concept" shows up more than "procedure" then they conclude curriculum most be poor and unbalanced.
sleeper2345 2 years ago
Are you serious? Who does this? For each book they do that? And that's the sum total of the thinking?
jamesblackburnlynch 2 years ago
This is done for the entire K-12 curriculum goals. If too many concepts are identified, over procedures, then we need to reduce the number of concepts. After all, didn't the National Math Panel say there needs to be a balance? This is the reasoning of lay people (i.e., school board members). After all, what knowledge do they have of K-12 mathematics?
sleeper2345 2 years ago
Well, if they were responsible, I would hope they would gather knowledge about K-12 mathematics.
And again, how do you account for all the school boards that have chosen EM? The propaganda machines are pretty loud that it is weak in procedure.
jamesblackburnlynch 2 years ago
The difficulty lies in the mathematical knowledge of the BOE and adults in the general populace. Unfortunately, there are many websites that spin the "back to traditional algorithms" web. It's very successful because this is what mathematics is to most adults. Perhaps we need more research on the sorry stated of adult mathematical knowledge to drive change. As it is now, too many adults seems to believe that in the "golden era" of the 60s and 70s that so much more math was learned.
sleeper2345 2 years ago
What impact will websites such as wolfram alpha have on the teaching of mathematics? It will do about procedure learned in K-12 and many that are the focus of calculus classes in college. How much time should be spent on procedures when such technology is available? It seems like students could learn much more and better mathematics if we did not spend so much time learning procedures.
sleeper2345 2 years ago
I don't think everybody should go to college. It's now where Universities are nothing more than an extension of high school & a business. Ppl just think it's something you do. Too many don't learn how to think & are nothing more than a blank disc & just try to repeat what's been put on it, thinking cannot be taught. Life itself should make one calculate. U.S. society itself with the way it functions from home to social situations is set up to produce idiocy on a mass scale.
asherasator 2 years ago
I have worked with students in the classroom for the last four year for 3 to 5 hours a week. I can tell you students don't find them intuitive. Since you agree they must ultimately learn an algorithm, why teach a highly inefficient algorithm. When they make a mistake, which happens frequently given the multiple steps, it is too many steps for them to find their error.
Why do you have to make videos teaching the method if it is just so apparent to students?
teamcasazza 2 years ago
First of all, you misunderstood what I said. I made videos about Elementary Math: what's in those books and why. I didn't attempt to explain how the different algorithms work. I wouldn't mind doing that actually...books don't explain themselves completely, but I didn't do that.
Secondly, you and I are having a fundamental disagreement about what we want to teach. I want understanding first and preeminently. Mastery of skill second and subordinately.
jamesblackburnlynch 2 years ago
So, the difference we are having is causing our different points of view. I don't think of the algorithms in terms, simply, of "efficiency." I think of them in terms of understanding. I want students to understand them. The partial products is the easiest one to understand (no hidden meanings in the carrying of digits). The others are used to push students to think about how algorithms are built. It's all about understanding.
Later, the mastery comes. I don't care which algorithm.
jamesblackburnlynch 2 years ago
The math part of the swedish equivalent of the SAT is done in an interesting way. You are given a problem. Then you are given two additional facts 1 and 2. Your task is to figure out if the problem can be solved using
A. (1) but not (2)
B. (2) but not (1)
C. (1) and (2) together
D. (1) and (2) each by themself
E. Can't be solved using the two facts
That is in my opinion an excellent way to test understanding of math.
wildclaw2 2 years ago
That does sound interesting. Is there a website with an example problem or two I could see?
jamesblackburnlynch 2 years ago
Here is a short example that I found. It is a pretty simple one, but it should show the basic idea.
Kim leaves city A at 08:20 and rides a bike with constant speed to city B, a distance of 24 km. When does Kim arrive at city B?
(1) Between city A and B, Kim maintains a speed of 18km/h
(2) Kim rides the first 12 km in 40 minutes.
The answer for this specific problem is (D) as You can use (1) or (2) by themself to solve the problem.
wildclaw2 2 years ago
And here is another one.
In a shop, flour was sold in packages of 2kg, 5kg or 8kg. The difference between the cheapest and most expensive price/kg was $2. Which of the packages had the lowest price/kg?
(1) Buying 2 packages of 8kg cost $4 more than buying 2 packages of 5kg plus 3 packages of 2kg.
(2) The 2kg package had the highest price/kg.
The answer is that you need both (1) and (2) to solve it. So C is the answer.
wildclaw2 2 years ago
Those are good problems. I do like the set-up. How many of these problems in what time limit?
jamesblackburnlynch 2 years ago
The math part of the test is 22 problems with a 50 minute limit.
wildclaw2 2 years ago
Everyday Math teaches as many algorithms as classic math, it just teaches highly inefficient algorithms. Whereas classic math maybe takes two steps, the Everyday Math algorithm takes 5, 6 or even more. This does NOT make math more intuitive. It makes math longer, more difficult, more error-prone and more confusing. Everday Math fails to teach math logic or basic math skills. It simply fail students. I recommend you look at the textbooks before you defend them. You might learn something.
teamcasazza 2 years ago
You are watching a video that is over two years old. Not only have I actually read those books, I've made videos detailing the materials inside them.
The partial products algorithm, in particular, is in my opinion, everything you claim it is not. It so clear and intuitive that many people come up with it themselves, and don't bother with the "standard algorithm" unless the numbers are very large.
And, in the end, I believe the students must master one of the algorithms.
jamesblackburnlynch 2 years ago
(. . .)
I'm just kidding about that last statement. But overall I'd much rather have them ask "What does this have to do with math?" instead of "What does this have to do with ANYTHING?" Most textbooks don't do much to alleviate the latter—this is very hard for a textbook to do, quite frankly. Any one that manages to do so gets my kudos--in my book, they have accomplished something very big!
TAStidham 2 years ago
I'm not sure I understand the distinction. Everyday Math is definitely focused on the applications of math. I'm not sure what the other option would mean.
jamesblackburnlynch 2 years ago
(. . .)
But if they happen to ask this question ("What does this have to do with math?") on a day when I am not particularly prepared, I just tell them to shut up and have fun with it.
TAStidham 2 years ago
That's too bad. Many students will be completely turned off by that answer and conclude that whatever math they are doing that day has no reason at all. For most people (sadly) math is not fun. Even when they understand it, few people actually really enjoy doing math. I'd have to say that most math majors I know don't seem to truly enjoy math.
I sure do. And I've met many who do too. But most, nope.
jamesblackburnlynch 2 years ago
(. . .)
It is a big victory when a student asks "What does this have to do with MATH?" It tells me right away that they have already accepted that the given scenario does indeed deal with something outside that world of meaningless numbers and manipulations. Confident that the proper content will emerge, I can just smile and ask for patience (hopefully this patience is earned as the course progresses).
TAStidham 2 years ago
Hmm....I'm fairly certain I'm not getting what you are saying here. When do they ask that? When are you talking about something that doesn't appear to be about math? Is this when students are confusing arithmetic with math? That is, if it isn't computation or arithmetic, it's not math?
jamesblackburnlynch 2 years ago
I'll give you an example. One assignment I gave last year was to fill out an 8-question "Which do you like better?" with two possible choices per question: Coke or Pepsi? Pancakes of Waffles? Wii or Xbox? etc. Initially this assignment appears to have nothing to do with math, prompting some to wonder why they are doing it. At this point, they do not yet know that they are going to translate each of their responses into 1's and 0's and learn about the binary number system.
TAStidham 2 years ago
A project like this is likely to elicit a "What does this have to do with math?" response at the outset. They wonder how their opinions about sports and candy could be related to anything mathematical. When I hear a question like this I know I have already grabbed their attention to some degree. However, when I hear the other question, "What does this have to do with MY LIFE?" this tells me that they are whiny and bored. I most definitely prefer the former.
TAStidham 2 years ago
(. . .)
As a high school teacher I strive to teach my students the skills the state thinks they should know. But even more I want the reasons for those skills to be tightly intertwined in the delivery. I try not to give my students any opportunity to question "What does this have to with anything real?" If a student lacks motivation (as many still will), he or she should be forced into admitting laziness rather than being able to pin it on "irrelevance" of the tasks they are assigned.
TAStidham 2 years ago
Agreed. I find that students are most likely to say "when will I ever use this in my life" when they don't understand the topic. No one asks me that when they get it. That comes up as a defense for something difficult.
jamesblackburnlynch 2 years ago
(. . .)
In real life math is hardly ever handed to you with pre-constructed problems. A meteorologist I'm sure can attest to this. In real life it is you who defines the problem, you who wrestles with it. Real life math in many cases DOES look like geography on the surface (or science, or humanities, or economics, etc.) It does not make sense to me how someone in a math-related field, who works with maps every day, can see a few maps in a textbook and assert immediately "This is not math."
TAStidham 2 years ago
100% agreement on that. It's as if she didn't actually look at the book there. It's all graphs and percents and fractions and good stuff. And all related to this one topic (the world tour) so that the teacher can keep coming back to the same basic application with many different approaches. There's even some meteorology in there.
jamesblackburnlynch 2 years ago
(continued)
The meteorologist, who likely had had an affinity for math all her life, denounced 15-25 pages of the textbook asking "Where's the math?" as she flipped through them quickly. Why? Because they were colorful. They were interesting to look at. They had maps--"that's not math, that's geography!" She did not bother to read or explain the activities associated with those maps, only that the pages were not "chuck full" of written exercises and practice problems.
TAStidham 2 years ago
Yep, agree again. This is what I meant above. Many people think if it's not straight computation it can't be math. They miss the whole point of what math is. And that computation is the final part of a math problem, and is usually the easiest part that a computer could handle, in real life. But the actual person has to get the problem to the point where a computer can handle it. That's the real math.
jamesblackburnlynch 2 years ago
(continued)
If math really is everywhere, like we want our students to believe, then a textbook that aims to provide a rich, meaningful context for students to practice real math must certainly be a step in the right direction. I can't say I vouch for throwing the traditional algorithms out the window, but I do disagree with the idea of discrediting a resource on the basis that it looks so different.
TAStidham 2 years ago
And the book doesn't throw the standard algorithm out. It eventually concludes that it is the one to use, for most people. But it works up to it, and demands that students understand it. One of my favorite parts of the book is that when it brings up the lattice method, it doesn't explain it to them. It shows an example and asks them to figure it out. Any student who can definitely understands the standard algorithm. Very challenging question.
jamesblackburnlynch 2 years ago
(continued)
Is math really its own separate world of thought, void of apparent meaning, that some get and some just don't? Of course not! I can't imagine any educator believing this--but the problem is that many students do--the majority most likely.
TAStidham 2 years ago
I agree that imparting a love of math to many is difficult; and I believe it is downright impossible with the approach used in most classrooms: "Math is what it is--working with numbers on paper. You may love it or hate it, but either way you need to be good at it." Hard to get people to care about math with this mindset. Of all the disciplines, it is perhaps the least challenged by adults in terms of its necessity in education--but also the least understood by students as to why.
TAStidham 2 years ago
Hello. I've been working as a math tutor (elementary algebra through calculus 1) at my college for several years, so I also get a lot of first-hand experience in how unprepared for college-level math some students can be.
I love math because I "get" math -- everything in the universe boils down to math. However I find it hard to tutor some students because I find it difficult to impart that intrinsic understanding of math. Maybe it's because of the way I was taught in school.
Uejji 2 years ago
I was taught to parrot multiplication tables and apply them to multiplication and division like anyone else my age and older was. I personally see it as kind of a shame when a student has to use a calculator for while I consider a simple multiplication such as 6x4. Maybe I expect the student to have memorized it like I did, or maybe I see the calculator as a time-waster rather than a time-saver for simple arithmetic, and I always teach my students that speed is important second to accuracy.
Uejji 2 years ago
As far as formulas go, a sad truth about math is that there *are* a lot of formulas to memorize (though I don't have to tell you that) so you can never get away from memorization. The Pythagorean Theorem for example -- I can understand right triangles and how one angle is 90 degrees and the other two are complementary, but if I don't know a^2 + b^2 = c^2 and that c is the hypotenuse, I won't be able to tell you the missing side of that triangle.
Uejji 2 years ago
The Pythagorean Theorem gets such a big name because it is such a significant result. The proof is not just common sense. I agree it does have to be memorized in the end. But I don't agree that there is a lot to memorize. Well, not compared with what people think.
For example, I got into a conversation here on youtube with a student. She believed she was being asked to memorize what 21*56 was! It's hard for students to distinguish what is to be memorized and what is to be figured out.
jamesblackburnlynch 2 years ago
I agree with you 100% that students need to be imparted this distinguishment, but it is a bit frustrating to see students trying to memorize things that cannot be easily explained. For example, I can't prove to you that pi*r^2 is the area of a circle without deriving it using limits, but the students that would understand that derivation would not need to be proven that pi*r^2 is the area of a circle.
So regardless of your level of understanding in math, there is a lot of "this is how it is."
Uejji 2 years ago
Yes, pir^2 is another one that is difficult to explain. Again, this example is from geometry.
But these are exceptions. I'm teaching College Algebra right now and there is virtually nothing I need to make them memorize.
The quadratic formula...I derived this for them using completing the square and so on...but I told them that this is the rare time I think it's better just to memorize. The derviation is too difficult to redo all the time. And it's not very revealing. But what else?
jamesblackburnlynch 2 years ago
Lately I've been working with summer students on linear equations.
The standard and general forms are probably the most confusing because it is hard to parse anything meaningful from them without plugging in an x to get a y and plot a few points.
Whereas slope-intercept form you can plot several points right away just by knowing the y-intercept and slope. Point-slope form is also meaningful because it still gives you the slope and a starting point.
Uejji 2 years ago
I agree completely. I always tell my students that the standard form is basically useless. It hides the slope and the y-interecept or any other point. The slope-intercept form is the one that makes the most intuitive sense (if you make sure they first understand that lines are models that describe a constant growth/decline by addition/subtraction). I tell them that I don't bother memorizing point-slope but I see the advantage to it. But standard form? Pass. (Only for vertical lines.)
jamesblackburnlynch 2 years ago
After watching more of your videos and talking to you some I think I agree with you -- math education needs to change on a fundamental level. If mathematical competency were to improve, then so would competency in every other subject that directly involves math: physics, chemistry, astronomy, etc.
However it is hard to change an education system. The most I hope to do is impart a love or at least an understanding of math to my students.
Uejji 2 years ago
Imparting an understanding is all we are after. Imparting a love of math would be nice, but it's not at all realistic to think you can do that with many. We can't really all love the same subjects. But the idea that students could imagine why someone would really enjoy math, even if they don't, would be nice. That basically can be passed on just by loving it and showing that love each day. And making sure that it makes sense.
Good luck with it!
jamesblackburnlynch 2 years ago
I know I really shouldn't post this because it's so stupid, but it is the truly correct answer to the person-cave question you asked...
They can't go all the way through the cave because caves have an end! It'd be a tunnel if they could go all the way through! That means it doesn't even matter the average height of the tunnel or how tall the person is! That's the real answer to your hypothetical question!
So, can I have a prize?
dexter111344 2 years ago
Do you really want to stick with that? That caves only have one entrance? I don't think that's a definition or anything.
From what I can gather a "tunnel" has to be intentionally dug. That's the difference. So caves can have an opening on the "other end."
Mathematicians are sticklers for defining your terms!
Your "no prize" is in the mail.
jamesblackburnlynch 2 years ago
I can't have the t-shirt you're wearing?! Well... What about one of your shoes? I just need one. Preferably your left, unless it's a slipper.
But I do agree with what you're saying about those books. They're trying to teach WHY an answer is what it is, and not just what the answer is. But the issue is that they don't bother to teach the easier way of solving equations once the children get the reasoning behind solving the questions. Instead, they just say to use a calculator.
dexter111344 2 years ago
If that really is so, that's not okay with me. As I said in my videos, I think the focus on understanding is essential, but I also think, that once the understanding is solid, then practice is essential too. Maybe calculators obviate the need for the standard algorithm (I don't think so, but let's just say they do), but the importance of practice is also a message that must be established early.
Talent is great. But talent and hard work, that's what it really takes.
jamesblackburnlynch 2 years ago
I agree completely with you.
dexter111344 2 years ago
Excellent video. Thanks.
I am also disturbed that most elementary school math programs put emphasis on memorizing specific techniques rather than on finding solutions through the use of solid reasoning. While I do think teachers should still teach the traditional multiplication and division algorithms, I have a problem with them in that it is nearly impossible to explain to an elementary school student WHY the traditional algorithms work. The reasons are more complicated than one might think
gotmilk1289 2 years ago
Even the average high school senior would have trouble understanding the derivation of long division. This is another reason teachers should teach some of the other methods mentioned in the video. They actually make sense!
gotmilk1289 2 years ago
I think the standard algorithm for multiplication isn't too complicated, particularly if one starts by looking at the partial-products method. The division algorithm is perhaps a bit much.
jamesblackburnlynch 2 years ago
Like many of these respondents, I wanted to thank you immensely for saying this. "An Inconvenient Truth" struck me very much as an argument for stagnation; primarily the message seemed to be one in which you teach what's familiar, even if it's known that what's being taught fails a great many people. I believe that rote mathematics can be useful for quick answers, but agree with you that rote does not teach the concepts behind mathematics, something which is sorely lacking in math education.
ytghanima 2 years ago
Please communicate the importance of reasoning with others. In Seattle, WA and other places, parents have banded together to disparage reasoning and thinking. They emphasize the use of multiple examples that students should mimic. They say that students should learn "concepts," but never describe what these are. Instead, the encourage moving back to traditional texts so that the "tried and true" will remain. I guess that means our jobs are safe. Few will learn math with deep understanding.
sleeper2345 2 years ago
I've tried to do that. My hope with this video was to have a dialogue with people about this issue. I have for a couple of years now. I doubt I've changed minds (nor has anyone really changed mine) but I've had some interesting discussions and I understand better why they believe what they do.
I did watch a Saxon video and they certainly do sound like they care about the concepts. The problem is their system doesn't really make the students learn it. They can succeed with pure memory.
jamesblackburnlynch 2 years ago
Just wanted to say that I think you have it exactly right. She isn't comprehending the larger picture either. The different methods are good for helping the student to 'understand' the math rather than just memorizing but the system on a whole must be one wy or the other. As you point out, if they are taught one way and then tested on another, everyone becomes confused. Glad to see an intelligent reply to her video.
JustThinkin13 2 years ago
Your video response was useful. I believe that students should be able to understand how math contributes to daily lives and how to utilize it. My friend told me how our teacher never tells us why we do some techniques. She only tells us that it is needed for our test. We agreed together that our teacher is not a superior math teacher. She only gives shortcuts to problems, but never the reasons why we do it. Maybe its because she only teaches high school, but that should not be an excuse.
cbnunya 2 years ago
when they get rid of "carry the one" it will be allot better.
heoTheo 2 years ago
Huh?
jamesblackburnlynch 2 years ago
I think what heoTheo meant was that even though carrying numbers in mathematical processes, namely multiplication and addition, it would be better to develop a way to avoid this type of algorithm especially when dealing with large numbers using only pencil and paper. unless heoTheo was just goofing around, in which case forget what i thought he meant hehehe
csnoopy 2 years ago
Well, if that's the case, then the cluster algorithm is close to what heoTheo wants. "Carrying" is hiding that you are adding a tens digit (or hundreds, etc.) and the cluster algorithm refuse to hide it.
jamesblackburnlynch 2 years ago
I just wanted to say thanks for making these response videos. You say exactly what I felt when I watched the 'inconvenient truth' video.
I'm actually a third year physics major, and I've thought on and off about going into education, but I've always been conflicted with the issue of perhaps being restrained into teaching to the test and such.
Anyways, thanks again for taking the time to make these responses.
alextheape 2 years ago
I found this kind of problem with physics, rather than basic math. I could grasp math easily, because I could work out how everything related. The problem I hit with physics is that suddenly I was spoon fed tons of equations, and while sure I understood many parts, it mostly seemed completely arbitrary and something that merely needed memorized. I found that it became far easier to understand when I understand the history of the discovery of the equations, but it was rare to get that.
Daniel092004 2 years ago
This is a different topic, but I think I understand what happens in physics. I took a physics class a few years ago to see what's going on there.
In physics, the mathematics of the students is always several years behind the physics. So, the physics teacher has to fake the math they use. For students with an intuitive grasp of physics, they can just ignore this, but for the more mathematical mind, it can drive the student crazy and prevent understanding.
jamesblackburnlynch 2 years ago
continued:
A lot of my teachers actually do show the derivations of the formulas, and then explain how we get there, but the problem is that nobody pays attention until the part where they get the final formula, and then they just go and memorize it.
How would you suggest we solve a problem like that?
PockyRulez 2 years ago
Yes, I call this the "blah blah blah answer" interpretation by students of a lecture. I've checked the notes of lots of students, and this is the norm. They don't bother writing any of the concepts/big ideas down. They think just the formulas and an example will do.
How do deal with this? Not easy, but the first step is to make sure that doesn't work. That is, test problems must be such that memorizing formulas won't get you there. They must challenge understanding.
jamesblackburnlynch 2 years ago
I think you're leaving out one factor: the actual students. Some students WON'T 'step back' and try to understand it. They just want to memorize the formulas and pass the test. It's very hard to make someone understand something when they simply don't want to, I've seen it in a lot of my classmates (I'm a grade 11 student in Canada).
I never memorize formulas for tests, I can't remember, I'll just derive it.
(continued in post 2)
PockyRulez 2 years ago
It's true that some just want to memorize. Or, really, many many do. And if it works, what's to stop them? It's important to show them that it doesn't work. I think the teacher actually has to have an honest conversation where students get to argue that shouldn't have to do anything more than memorize. And then the teacher should make their case that that isn't enough.
What's more, some students who want to learn more, don't really know how to. They want to...but don't know how.
jamesblackburnlynch 2 years ago
It depends at what age you get them I think. I think all children are naturally curious until it gets beaten out of them in the current teaching methods..
spleenblender 2 years ago
Thanks for your thought on this.
You say that the idea behind "these books" is to teach the idea behind the algorythm. But I don'tsee how these other methods teach the desired idea at all, either.
Her point on the original video is that the common algorithm is simpler and better. Neither teaches much of the idea.
I guess I am thinking you might be a little naive in assuming that "these books" are trying to teach the thinking. I see no sign of that.
Comments?
AOZXRAY 2 years ago
Have you seen my other videos where I discuss the books themselves (which I finally got my hands on)? There I detail specific ways the books get at the concepts.
For this particular one, the cluster algorithm, in my opinion, is the idea of the standard algorithm, but without the "mysterious" carry digit. It clarifies the whole procedure.
The lattice method is used in these books to push even further the understanding by coming up with a defense of a strange algorithm.
jamesblackburnlynch 2 years ago
I agree that word problems (or, to put it more simply, real-life problems) should be the goal. But, as you probably know, students hate word problems, usually. That's because word problems require thought. And we seem to keep telling our students that math should be free of thought.
jamesblackburnlynch 2 years ago
Is that a satisfying answer to the question? I think most of my students would find that answer very off-putting. To me it sounds basically like "you won't ever need this."
jamesblackburnlynch 2 years ago
I guess my question would be this...you talk a lot and make very valid points about those who just "pick things up" vs. those who don't. And that the program needs to work so that the ones who can't just pick it up automatically aren't having a disservice done to them. But is it doing that. I've recently helped a kid by teaching him a way to do fractions that he got. But his teacher told him he was doing it the wrong way (not her way). Even though it works all the time. How is that helping?
fairynanook 2 years ago
Clearly, that's not helping. But it's also against the philosophy of the book. Multiple approaches are more than welcomed, they are required. Now, it's possible the teacher was saying that the student needed to be able to do it the way you showed and the other way too. But if she told him her way was the only acceptable way, then clearly she doesn't believe in these books.
jamesblackburnlynch 2 years ago
But she is using the TERC method and she clearly believes in it. This child got the highest score on his latest test, after I taught him to understand in 10 minutes what his teacher couldn't teach in 2 weeks. And clearly, from the scores, the other kids didn't get it either. Perhaps this is just a bad math teacher, but I am not a math teacher at all, and yet I have taught this child fractions where his teacher failed. And he understands why with my method. And he gets the right answer.
fairynanook 2 years ago
Again, my question would be, if the majority learn "why" just by learning the formulas, and from my experience in college, most did, why change the method so that the minority might learn why, and instead no one learns at all. I get why it is important to understand the why. But I don't agree with the argument that the old methods don't work to teach this for the majority. And again, I am not a math teacher, so obviously my view is a bit off.
fairynanook 2 years ago