...just invert and multiply!

This seems to be one of the fundamental philosophical questions in math education. Do you teach tricks and rules, or do you let students explore and construct their own knowledge? Does it matter what type of students you are working with? Behaviorist, Constructivist, Socio-Constructivist?

I, personally, believe in Socio-Constructivism. The idea is that the teacher creates a well-structured pathway for learning that takes students through the levels of understanding. Students start off being given basic facts or information that they will need, and then are given time to work (individually or collaboratively) on expoloration or inquiry based activities. Through this process, they begin to generate conjectures and create a quasi-mathematical understanding. Then, through class discussion and direct instruction, the teacher helps correct misunderstandings and formalize the knowledge (i.e. algorithms, processes, etc.) The drawback to this, of course, is that it takes a lot more time - both to plan the materials, and actual class time. Is it worth it? That's the real question. I believe it is, but I think there are those who disagree.

One of the classs that I have developed is called Numeracy, in which we put freshmen who test below 7th grade level when they arrive (this is typically 70 - 80% of the class). We start off at basic operations and place value concepts - this takes the entire first semester. The second semester is all fractions. We spend a few weeks working with fraction circles, drawing fraction bars, using reasoning, etc. to compare, order, and evaluate fractions. We work on determining if a fraction is closer to 0, 1/2, or 1 whole. We figure out how you can compare 7/8 and 8/9 by reasoning. Then we spend 6 weeks adding and subtracting, with manipulatives, with pictures, and finally, with the algorithm. Then there are 6 weeks dedicated to multiplication and a few more for division. Yet at the end of all this, I still have a large number of students who haven't learned to work with fractions fluently. Sometimes, there is the temptation to teach the rules and then practice them to death, but in my heart I believe this won't work. Plus, if you blindly memorize the "flip and multiply" rule, you haven't really learned anything about division and you won't be able to apply that knowledge to other situations (i.e. Algebra). Also, if you develop no context for your algorithm, you have no way of knowing if what you are doing makes any sense.

I think students have been trained to want algorithms in math. They resist exploration. A noisy class will quiet down and get to work when a worksheet is put in front of them - why? Even if it is not being graded, they will rush to try and get answers down, regardless of whether they are learning or not. Like lemmings, they seem compelled to "finish the worksheet". It boggles the mind! "Mr. Greene, just tell us the easy way! Stop asking us questions!" How many times a day do I hear that in Numeracy? Students know that drawing fraction circles, for example, will help them solve a problem, but they would rather ask me for help, or just skip the question. When I force them to draw a picture, they begrudgingly do so, then look at their picture and say, "Oh, that's all you want us to do? That's easy!" and proceed to answer the question with little difficulty. Yet, on the next question, the process will repeat itself. Patience is definitely a learned skill!

I am happy to say that, at the end of Numeracy, most of the students have learned that, when adding or subtracting fractions, **you don't add across!**

Any thoughts on the matter?

What Do We Mean by ‘Understanding’?

1 day ago

## 31 comments:

I had lots of discovery work at school.

From a student's point of view - and I was one of the stronger students - they're confusing and exasperating if used too often. The odd Friday afternoon it's fine, but floundering around a lot is miserable.

Meanwhile I learnt a whole lot of algorithms and over the years began to understand the basis behind them.

I don't think you can force understanding. I used to often just memorise the explanation as to why something worked. Eg I know that when studying calculus the teacher carefully explained why we'd take the derivative and then set it equal to 0, and she drew graphs, and she had us draw graphs, and etc, but for a couple of years that was just a mysterious magic trick to me that setting the derivative equal to 0 found both the min and the max. I memorised the explanation and would regurgitate it on maths tests designed to test our understanding, but it was only a couple of years later that I understood why it worked.

Conceptual understanding comes when it comes.

Liz here from I Speak of Dreams. Most of my teaching has had to do with physical skills (riding horses, skiing) -- the constructist approach just would not work, and could be actively dangerous.

Skills have to be developed and practiced to the point of automaticity.

I like to use the metaphor of musical improvisation--say, jazz, on the piano. The musician must have some skills completely automatized -- say running the scales for the two octaves around middle C. So you can start improvising with just those notes. Thus you can "play" --the constructivist sense--before you have a complete set of skills.

In my view, a good teacher alternates between building skills, fostering discovery, and "playing the game" to the limit of the mastered skills. But the jump from the skills in hand to the discovery area needs to be quite small--less than a 10% jump, however you want to quantify that.

It is often difficult for the expert (the teacher) to really remember what it was like not to have mastery. I know I trip over this all the time. Something that is automatic to me is out of reach of the student.

I am wondering if your students may have incomplete automaticity in the basic math facts, which would interfere with their ability to form conjectures and explore.

Let's take another image: riding a bicycle. If your students aren''t bomb-proof on balance, so it takes place without conscious thought (this would be automaticity of the math facts), they are using up energy to balance, and can't turn smoothly or control the bike well (use concepts) for fear they'll fall off the math bike.

I completely agree that students need some exploration, and not just memorizing rules on a lot of topics. You can tell the topics that need this treatment because they are the topics (fractions is a classic example) where students can memorize the rules beatifully, and yet come back the next year and do it all totally wrong. It appears to take more time to guide them through a real exploration of the subject, but only because you're not counting the time next year's teacher will spend reteaching the same topic.

The objective of all this, I think, should be for students to develop an intuition that will make correct strategies seem natural. I don't think anyone yet knows the best way to do this (especially with kids who have learned it wrong the first time around), but there are lots of things that seem to be making progress (some people like to do percents first, and use those to explain fractions, for example). I for one would love to hear more about what you are doing with fractions, perhaps as next year progresses

I don't believe in discovery work at all--it's taken the greatest minds the human race has had to offer a few millenia to create the systems of math we use today. It seems foolish, and indeed hurtful, to expect students to create what it took geniuses to create. Heck, not a single genius in the Roman world had even developed a zero!

I believe in direct instruction with thorough explanations. I teach algorithms along with explanations of either why they work or how they were developed. "Invert and multiply" is a fun and easy one.

On the other hand, Sherman Stein, retired UC Davis math professor and author of several fantastic books, isn't a fan of invert and multiply. He suggests getting a common denominator and then dividing straight across, as we do when multiplying fractions. The denominators, now being the same, will always divide to one, so the quotient of the numerators is the answer to the original division problem. It has two benefits--one, students already are used to finding a common denominator from adding, and this new method is more intuitive than "invert and multiply".

Just an idea.

A common denominator is what is basic. To order the fractions.

Teaching them to reason why 7/8 is smaller than 8/9 by looking at 1/8 and 1/9 is fun and important, but it is no substitute for teaching kids that they can find out precisely whether 3/5 is bigger or smaller than 4/7... by renaming each fraction in terms of a common denomination of the whole interval. THAT is true mathematical power!

"I will not just sit and wait, I will redenominate."

Also see Wu's papers, for instance, http://math.berkeley.edu/~wu/six-topics1.pdf.

Darren,

I do think that an inquiry/discovery based approach can help students develop critical thinking and problem solving skills in a way that no direct instruction with examples class ever could. I also agree that this is not necessarily the best way to learn math and it is not reasonable to expect students to figure out everything on their own.

That's why I believe in a socio-constructivist approach. For example, for division of fractions, I first spend months developing the idea of what a fraction is, and how to compare, order, estimate, add, etc. To do this, I don't give students fractions and say, ok, go explore! I carefully scaffold the problems I pose to them, using word problems that model different ways in which an operation might be used, manipulatives, and mathematical explanations. As we move forward, I try to elicit emerging mathematical ideas, and I do give feedback, showing students that they have a misconception, telling them they are on the wrong track, arguing that the formula they "remember" from last year is not correct, providing further problems to test their conjectures.

Finally, when its time for division, I again start with situation problems, looking at both partitive and measurement problems, and various combinations of fraction / whole number for the dividend and divisor. Its always amazing to me to see how some students really latch on to some division concepts and totally can't see others. I spend the whole unit, again and again, explicitly tying our findings back to the idea of a reciprocal. In the last couple weeks, I do introduce the common denominator method for division (based off of drawing fraction bars first, and then formalizing it as an algorithm), and

finallyI get to the multiply by the reciprocal algorithm. The reason I do all this is because I find that different students seem to latch on to the different strategies. At first, I force all students to draw fraction pictures on their tests to explain their answers. As the unit progresses, I let them choose to do so or not. Some students keep on doing it, some stop right away, some do a mixture - they use algorithms for the problems that are immediately clear to them, and then draw pictures for the ones that are harder to understand. When I see this is when I feel that I have been most successful, because it shows me an advance in their critical thinking.Overall, I think we should teach algorithms and formulas; however, I think this can only be effective when students have a firm grounding in the underlying concepts. I think this became more clear to me after working with high school students who still don't know the fundamentals; my students who still don't really understand the base-10 system never master rounding, even when I try to just teach them the algorithm.. they just can't remember it or apply it correctly, even though it seems like a very simple algorithm.

To anonymous,

I agree that the reasoning I'm talking about is not a substitute for finding common denominators - its a supplement. I definitely teach both. However, I always start with the reasoning, because I think this kind of thinking is what successful mathematicians do.

If I asked you to compare 3/7 with 11/13, I highly doubt that you would bother to find common denominators. But my students would, if that was the only thing taught to them. One of the skills I work on a lot is estimating and determining if a fraction is closer to 0, 1/2, or 1 whole. My students would say that 11/13 is clearly bigger because it is much more than a half while 3/7 is less, or because they would picture the fractions in their minds. I think that this kind of thinking has to go hand-in-hand with algorithmic ability, to create students who can really reason mathematically.

algorithmic abilityFinding a common denomination is more than some unthinking, algorithmic ability -- it is the conceptual foundation of what a fraction really is -- which is to say, trying to get at all the points in between the whole numbers.

If I ask you, which is bigger, 7 dimes or 3 quarters? You probably mentally converted each quantity to pennies before answering...

The reasoning steps you give -- is it near one? is it less than a half? is it close to zero? --represent a short reasoning

algorithmthat breaks down for a large class of comparisons. I think THAT is what leaves students feeling unsure, and unsafe, even when you venture into precise calculations. You may be inadvertently causing your students to feel that mathematics is tricky rather than straightforward, that they should be "seeing" something they don't see yet.I respect your willingness to share your thoughts on instruction. Obviously, I'm hoping to influence you! I worry that constructivists, of any stripe, are spending far too much time worrying about students who first find the common denomination, in order to compare 3/7 and 11/13. I worry about teachers who subtly ridicule children for "not seeing" that 3/7 is "close to zero" and 11/13 is "close to one".

I think that constructivists worship the ability to estimate, which is a high order skill, and which for many of us mere mortals, comes a long time

aftera lot of precise calculations. And this contributes to the problem of making children gun-shy about math, even though that's thelastthing in the world constructivists mean to do. It's an unintended consequence.From my perspective, "just seeing" that 3/7 is way less than 11/13 is supposed to be the

frostingon the cake after a child has spent a lot of time with worksheets filled with fractions.With the children I tutor in math, I start with precise definitions followed by calculations, and I end with the estimating.

Thanks for letting me share my 2cents.

Teaching facts first, and then challenging students to use the facts in a new way they haven't seen before is what I think Dan is after.

I find that some students can make the leaps, others have more difficulty. I offer hints, ask questions to get them to make the leaps. If it doesn't work, I just explain it using the facts they know. It sinks in. The mistake is in thinking that "Gee if they don't make the leap by themselves, it's just rote learning and they're not learning". That's not the case.

The fact the kids like to do a worksheet is not a bad thing. Automaticity is something you want, so they don't waste time hung up on trying to rememberwhat 8 x 7 is when they're in an algebra class and you're trying to teach a more sophisticated concept.

Division by fractions is not done overnight. In the Singapore math series they start out with multiplication by fractions, learning that 1/2 x 6 is the same as 6 divided by 2, so the inversion is taking place intuitively.

There are successive steps : division of a fraction by a whole number (1/2 divided by 4 is the same thing as 1/4 of 1/2 which is 1/4 x 1/2) until finally they get to a whole number divided by something like 3/4.

There are various ways to teach why invert and multiply works. I taught my daughter and her friend using Singapore this year. Her friend understood why. My daughter kind of did. There's no sense kicking yourself and saying "Oh, she's doomed, she doesn't REALLY understand". You want them to see the pattern. You want them to be able to operate with it. And most importantly to know what it means to divide 3/4 by 2/13.

If they can follow the explanation, great. If not, they'll get it when they get to algebra. Like I did.

"Critical thinking" and "problem solving skills" are best developed with by a mastery of facts. Students cannot think critically about what they do not know.

"Guiding" students to deeper understanding is obviously good pedagogy and good direct instruction.

If you interpreted my comments about direct instruction to mean "Here's a problem, here's how you solve it" then I obviously failed to communicate clearly.

"If they can follow the explanation, great. If not, they'll get it when they get to algebra. Like I did."

But here exactly is the problem - my students have a history of failure in math. Most of them have not mastered the fraction algorithms, let alone understanding why they work. They have almost all gone through algebra already, and they still haven't learned. So now it is our job to teach them algebra, as well as trying to reteach the entire concept of fractions, decimals, and percents which is something that really needs multiple years to master.

My purpose for starting this blog is to get ideas out there. Though I clearly have my own philosophy right now, it's not something set in stone, and I am eager to hear other people's opinions and reasoning.

My outlook on teaching is integrated with the specific type of students I work with, as they are the only ones I have ever taught. My experience is that my students have been taught the fraction algorithms year after year, yet they still can't do them when they reach 9th grade. So why is this? I know there is a variety of reasons, not all related to their schooling. My experience has also shown me that students who begin to understand why algorithms work, instead of just memorizing the algorithm, are better able to both retain the algorithm, and correctly apply it to problem situations later on.

Just to clarify: to me, rote learning is, "Hey kids. To divide fractions, first you flip the second fraction and then multiply across on top and bottom. Ok, let's practice 100 problems and then take a time-test". I fully agree that there are (many) times when I need to fill in the gaps for students and lead them from one step to the next. That's why I don't call myself a constructivist. I don't believe that students should figure everything out for themselves, by themselves. But I also believe that a student should be given opportunities to explore carefully crafted problems that will lead to the understanding/development of an algorithm, before the algorithm is explicitly taught, and that students should not be assessed solely on use of the algorithm.

If I asked you to compare 3/7 with 11/13, I highly doubt that you would bother to find common denominators... One of the skills I work on a lot is estimating and determining if a fraction is closer to 0, 1/2, or 1 whole. My students would say that 11/13 is clearly bigger because it is much more than a half while 3/7 is less, or because they would picture the fractions in their minds.With that carefully contrived example, it works. But what happens once you step away from those somewhat obvious numbers? For example, compare 11/13 and 16/19. Clearly, both fractions are closer to 1 whole. But how do you know which is bigger without converting to a common denominator?

Just to clarify: to me, rote learning is, "Hey kids. To divide fractions, first you flip the second fraction and then multiply across on top and bottom. Ok, let's practice 100 problems and then take a time-test".Well, a better method of rote learning

shouldlook something like this:"Hey kids. To divide fractions, first you flip the second fraction and then multiply across on the top and bottom.

The reason for this is because division is very similar to the multiplication we did earlier. You see, [insert explanation of how it relates to multiplication and review the rules of multiplication of fractions here].Ok, let's practice 100 problems and then take a time-test".Self-exploration can be a fun exercise, but a better approach based on my past teaching experience is to simply give a good explanation of

whythings happen that way. "This is why we do X. See how it related to Y, in our last unit?" Otherwise, you end up with half of your class endlessly noodling about, while the other half blazes forward.Dan,

The problem may be more than just not understanding the "why" behind the fraction division algorithm. It may be in not knowing what it means when you divide fractions; i.e., what is a story problem that goes with 1/2 divided by 2/7? What is it we're actually finding out? Many kids have a tough time with this, and some of this relates to not getting practice with the word problems that go with the algorithm.

The second issue may be in making the leap from numbers to letter symbols. I don't know what your students' backgrounds are, what they've had in the lower grades, etc. So it's a little hard to provide you with guidance. I've just worked with sixth graders exposed to division by fractions for the first time. In some of the exercises we did there was some representation of word problems in symbols (i.e., solve for x type problems).

It is sounding to me that explaining the derivation may be part of the problem if they learned these algorithms in isolation to how they're applied, or how division of fractions relates to multiplication etc. But it also sounds like they may not have experience in applying it.

Liz here from I Speak of Dreams--I just ran across this from Michael Drout, who teaches English at Wheaton College (the Massachussetts one, not the religious one). He is talking about teaching writing, but I think it applies to all fields of intellectual endeavor:

Approaches to Composition

"It is a cognitive problem: have a student write a 'what I did on my summer vacation' essay and it will come out grammatically clean. Have the same student write an essay about a difficult topic and all of sudden there will be subject/verb agreement errors or comma splices or misplaced modifiers. I wish faculty would realize that not all of these problems are due to failures in instruction or student laziness: a good prorportion of them are caused by the students not being able to keep all the cognitive balls in the air at the same time: thinking about a difficult topic, constructing an argument, writing clearly and grammatically."

I don't think direct instruction is going to be useful in this situation. I agree that there are good and bad ways to do direct instruction, but these students are in the numeracy class because they had the direct instruction experience and it didn't take (the 30% for whom explaining the algorithm and assigning practice was effective (as opposed to asking them to figure it out/understand it/apply it/explain it) aren't in this class).

If you want students to put more thinking (and not just more time) into learning fractions, you need to change the problems and the expectations to ask more of them than the last time around. I think a semi-discovery technique does a lot of that. A variety of word problems (especially if they are mixed up--not just a whole page of division word problems) helps with that. "How People Learn" says that one of the things that helps with transfer of knowledge from one situation to another is to solve problems in a variety of contexts (this doesn't mean different word problems, it means: worksheets, pictures, word problems...). I don't know a middle/high school teacher who isn't frustrated with how little his/her students know fractions, so if Dan's strategy works (ie. the students aren't making the same mistake next year) I think lots of people will be interested in this.

Of course 3/7 and 11/13 is a contrived example - but so is everything that we teach. We always contrive our examples and word problems and equations to illustrate whatever point we are trying to make.

I do make sure to teach the common denominator algorithm - in fact, its the end goal of the unit. But the way I apporach developing concepts is by thinking about how a numerate person would approach a given situation. When asked to compare fractions, I believe a numerate person would first do a quick analysis. What that analysis is depends on the exact problem and what that person is most comfortable with, but it would probably involve some estimation, visualization, fact recall, proportional reasoning, etc. If nothing seems to kick in, then the numerate person would last reach for the algorithm (this is assuming that the algorithm is more time consuming or complicated than the other methods). This is how I want my students to approach their math problems, because I think it will help them be more successful.

I want to teach kids multiple ways to reason through a situation, to help them put together the dots, and to develop a more holistic way of thinking about math. Right now, my students see math as a bunch of discrete problems with a bunch of mysterious steps that have to be followed. I know I can't expect them to see the whole forest right now, but I'm trying to move them in that direction.

The problem may be more than just not understanding the "why" behind the fraction division algorithm. It may be in not knowing what it means when you divide fractions; i.e., what is a story problem that goes with 1/2 divided by 2/7? What is it we're actually finding out?I completely agree with this. That's why I start each new fraction unit (comparing, adding, multiplying, dividing) in that very manner. I give students a pre-test diagnostic that includes half story problems (with a variety of mathematical structures) and half non-story problems. Students generally score in the 20-30% range on these pretests.

Then, I move into the unit by with several lessons dedicated to working on story problems, coaching kids to solve the problems by using fraction circles, drawing pictures, and using any other verbal or mathematical reasoning that they want. As they go, they begin to see "easier" ways of solving the problems. Then, we start formalizing what they are doing, and I start moving them towards the algorithm. Next, I move them on to non-story problems, where they continue to use the manipulatives and pictures methods. Finally, I use direct instruction to make sure everyone understands what the algorithm is, and we practice applying it to the problems. When I give them the post-test, I test them equally on the story and non-story problems. I see growth in both areas, but, interestingly, I tend to see much more growth in the story problem section (not just more growth, but higher absolute results). I'm not sure why that is, but I do have some ideas.

Discovery is a waste of time for kids who didn't understand algebra the first time. Some people just need to memorize the rules.

However, I've taught a number of people who are comfortable with fractions as constants that blank out when presented with fractions using variables. You might want to do more work with that.

Also, I've yet to see a kid who doesn't instantly grasp the formula (if not the concept) of fraction addition:

a/b + c/d = (ad + cb)/bd

or, for division:

a/b * c/d = a/b * d/c

So if you are using verbal explanations and examples instead of a universal formula, you might want to give it a try.

One other thing I notice is that kids learning fractions often confuse addition and multiplication and forget which is which.

I use this example: Taking a group of something (pencils, erasers, markers, whatever), I ask the kids to add half.

Add half of what? they always ask.

EXACTLY.

Now, take half of the group. Which, of course, is easy.

So that's how to remember which is which. When you "add half", you need to know "of what", which is why you have to find the common denominator. But you can "take half" of anything, so you can multiply without any extra steps.

Discovery is a waste of time for kids who didn't understand algebra the first time. Some people just need to memorize the rules.How do you know this? I have many students who were taught the rules only, with no chance to explore those rules, and they don't remember them at all. I agree that some students can learn math by memorizing algorithms and practicing them, but the majority of my students can't. I have tried this with certain concepts, just to see. For example, in Algebra 1, we have the "perpendicular lines have negative reciprocal slopes" standard. For the first few years, we just gave them this definition, explained why it worked, reviewed reciprocals, and had them do practice problems. This was an utter failure. Students just couldn't do even the most basic problems. I remember, at one point, having just reviewed the process. The notes were still on the board, there was a poster on the wall, and there were examples on the paper in front of them - with all of that, I still couldn't get many students to do a problem right, or even explain to me what the process should be.

With students who have failed math a lot, this type of block occurs often. It is frustrating, but at the same time, it really pushes me to think through a concept, and analyze why students aren't learning. With this, and with other problems I noticed in their work with linear functions, I realized that their weak understanding of slope was the cause.

So, for the following year, I reworked the unit to spend a couple of weeks really looking at slope - and not just the slope formula. We started with verbal descriptions of physical situations (i.e. people travelling, amount of money in an account changing, water level in the bath tub, etc.) and asked students to come up with visual representations. As a class, we then analyzed representations students did - what made them effective/ineffective. Then, we showed how a linear graph could be used to convey the information most efficiently.

From there, we spent a lot of time looking at graphs without number markings, asking students to reason, for example, which person was running faster and to explain why (i.e. she travelled the same distance in a shorter amount of time, so she must be going faster). This greatly helped students build their understanding of linear graphs - they could see why a steeper line was faster motion, or a greater rate of change, and why a flat line showed no motion (oh, time is passing but they are not changing where they are!). From there, we went on to including the scale markings, having students precisely calculate rates of change from the graph, and then finally introduced the slope formula.

Then, I continued on with the development of the linear function equations and the standard problems that go along with that. It took a lot of time, but I think the payoff was worth it. Instead of spending the same amount of time reviewing and reteaching the formulas (which is what I had done previously), students actually remembered and were able to use the formulas, much more so than in years past. This deeper understanding of slope helped them make more sense out of parallel and perpendicular lines and why their slopes act as they do.

But the way I approach developing concepts is by thinking about how a numerate person would approach a given situation. When asked to compare fractions, I believe a numerate person would first do a quick analysis.I disagree strongly. Novices need to start in a different place and do things differently, at first, than an expert does in his or her full wisdom.

I am a numerate person. Yes, I would do a quick analysis first. But I did not do a quick analysis when I was ten years old and learning fractions.

There is a lot of research showing that successful readers move from decoding words by letters or parts, to recognizing whole words by sight. Whether they are self-taught or instructed. You can't teach kids to read by coaching them to recognize whole words by sight -- to imitate what an expert does. They have to begin at the beginning.

Beginning mathematicians don't become experts just by putting on a coat and tie and acting like experts. There are foundational experiences in calculation that support individual movement towards expertise.

There is no royal road.

These kids have to put one foot in front of the other.

Let me say that it is fantastic that you are both examining your own teaching, and you are willing to put it out here for discussion.

I do make sure to teach the common denominator algorithm - in fact, its the end goal of the unit.I've read all of your responses to my comments and the other named and anonymous commentors, but I maintain that

finding common denominators is not an algorithm, it is a conceptual approach.Of course 3/7 and 11/13 is a contrived example - but so is everything that we teach.I'm a different anonymous and I didn't call your example contrived -- please don't dismiss my earlier comments or this comment.

Speaking of beginning steps,

It's very important for kids to learn to handle fraction, in isolation or in story problems,

with denominators other than the "friendly" ones: halves, eighths, tenths.It doesn't sound like you are making that mistake, but don't neglect that having students deal with the strangeness of sevenths, elevenths, and thirteenths may be a necessary intermediate step on their way to dealing with x's, y's, and z's.As my "Unexpected Mathematician" Terry, the auto mechanic said, "Math is different from all other subjects; you can't just memorize it and expect to understand anything." I agree - there's a time for concept building (at initial introduction) and a time for memorization (when the concept is used as tool). But, students who memorize their way through K-12 but don't understand anything are at a great disadvantage in college.

" I have many students who were taught the rules only, with no chance to explore those rules, and they don't remember them at all."

Your students weren't "taught" the rules, though. As you say, they know very little algebra at all, despite having sat through a year of it. So there's no evidence they ever learned it in the first place.

It's far more likely they never learned the rules than that they learned them and forgot them.

I have to wonder again, though, why your school is pretending that the students can learn calculus by their senior year when, at the end of their ninth grade year (or is it eighth?) they've not mastered fractions or basic algebra.

You need two weeks to explain the concept and principles of slopes to a group of kids who purportedly already passed algebra, and yet want to teach them geometry in five weeks over the summer. This seems profoundly unrealistic.

Oops, I just saw your response to me in the other thread--I should have checked that first. So I take back the question about calculus.

However, from your response, it appears that 50-60% of your students fail Algebra the first time at the school? That's rather shocking. Is that with or without the "discovery" work?

I'm sure you know of the data showing that passing algebra is a key educational success indicator. Many people seem to think this means that students should take algebra. But it seems to me it's more an indicator of capability--that if you have the capacity to understand algebra, you have the abstraction skills to succeed in higher education. Failing algebra, then, could indicate fewer abstraction skills.

If that's the case, then "discovery" work focusing on the big picture behind the math would probably be a bad approach to teaching kids who didn't grasp algebra the first time.

Cal,

The failure rates were shocking to me too, when I first started. No matter what I tried, and how hard I worked, I had so many kids that were failing. But soon, we started to realize that there was a direct correlation between students failing Algebra and their basic numeracy ability. This shouldn't come as any surprise, of course, but our data clearly showed that students who were able to increase their ability to work with basic operations and fractions were able to succeed at algebra.

When I say students "passed" algebra in middle school, it's important to take into consideration the math education that was being given to them. Many of the students I work with have gotten passed from level to level when they shouldn't have. They reach 8th grade without having developed strong numeracy skills (for a variety of reasons) but then the district requires them to have Algebra 1 as 8th graders, even if they are not ready (in an effort to leave no child behind). If they truly assessed their students' ability when assigning grades, I highly doubt that they would have as many passes as they do - how else can you explain our students who have passed algebra, but don't know how to solve a basic equation?

At our school, we require students to pass Algebra 1, 2, and Geometry to graduate, because those are the requirements to get in to a CSU/UC school. Of course this is a stretch for many students, but the mission of DCP is to take students who are underachieving, and first in their family to go to college, and send them to a 4-year university. We are a charter school, so only students and families who want this (knowing full well the requirements that will be asked of them) sign up - and each year, our waiting list grows. We do a lot of things to help them in this effort - longer days, smaller classes, mandatory tutorial, bringing in comunity tutors, parent involvement, dedicated teachers, etc. I have seen many students over the years go from knowing very little to making huge successes. We've graduated three classes already and have students finishing their sophomore years in college. Of course, we have had many students who have not made it to graduation too. We have been working on our program and trying to make it better, but our student poplulation has a lot of factors working against them.

There are a few students each year who have true special needs or learning disabilities who we have not been able to serve well. Others decide that the amount of work we demand is not for them. Some, though underserved in the past, have amazing minds and immediately begin to flourish under the high expecations that are set. For the rest, we see slow, steady growth through sheer determination - student fail, repeat, learn, grow, fail again, and finally pull it together.

Many student who fail Algebra 1 the first time eventually pass and then make it all the way through. At our school, in a way, we celebrate failure, and we beleive that abstraction is just another skill that can be coached. We don't give out D grades (which do count as passing at other schools) because CSU/UCs don't count them as passing - and this makes our failure rate higher. But when kids fail, we provide them many opportunities to learn from their mistakes and have another chance. We hold the line quite rigidly, though, and many students end up repeating a year due to failing a required class. Many of our students get on the "5-year plan" - and that's fine, because they have so much remediation that needs to be done.

In the end, we recognize that many of our students will not become mathematicians. They have had too many years with no foundation built. They will go on to study the humanities most likely - and that's fine. Their dream is to go to college and be successful, and they will.

This ties back to the calculus question - for those students who do have higher math skills, we want to make sure they are able to take the classes they need so that they can compete in college and take majors like engineering or math if they choose.

I disagree strongly. Novices need to start in a different place and do things differently, at first, than an expert does in his or her full wisdom.

I am a numerate person. Yes, I would do a quick analysis first. But I did not do a quick analysis when I was ten years old and learning fractions.

You make a good point here. But isn't it possible to scaffold the analytical skills we are talking about? In other words, I don't start off expecting students to be able to quickly estimate when comparing fractions - I try to lead them to this over a period of weeks by teaching them specific ways to look at what a fraction is.

I think that you can look at common denominators as both a process and a conceptual approach. To me, it is conceptual only if you understand that common denominators means you are finding equivalent fractions with the same size pieces. It is "algorithmic" if you only know the process but don't know how it relates to what a fraction is. I think "full understanding" is when you understand both the conceptualization

andthe algorithm.My contention is that the students I work with tend to need a conceptual background in order for the algorithm to stick. So that's why I have them add with manipulatives first (to see that you can only add same-sized pieces), then move that to pictures, and then, many of them come up with the mathematical algorithm on their own. When I give the algorithm as a lesson in class, for those who don't immediately understand it, they are more able to see why the process works when I can refer back to the pictures that they understand.

Thanks for your comments! I've tried to examine my own teaching since year 1, and it can be quite hard to do, because you get locked into a pattern of thought. That's one of the reasons I wanted to start this blog and hear ideas from others.

I don't think failing algebra should be celebrated. It's got to be hard on the kids.

While I certainly agree that incompetent teachers haven't helped these kids, I don't think we should dismiss the significance of failing algebra again, after being in your class. Failing algebra, as opposed to not being taught algebra, is a significant differentiator, and I think it's important to acknowledge that normal methods don't work.

After all, what is the difference between those who grasp algebra and those who don't? The first group grasps abstract concepts and the second group can't. I'm not convinced this can be taught, through discovery or any other method. But the difference is profound. Kids who grasp abstract concepts don't need to stop and think about how to add fractions using variables, or what the ratio of sides in a 30-60-90 right triangle are. They might not remember it in 10 years, but they easily understand that certain facts are basic and must be used, singly or in combination, to solve problems.

Kids who never mastered algebra or geometry take a long time to realize which facts they needed to solve the problem, and then even longer to draw out from memory what those facts are. This creates enormous problems in working math problems that require several different math facts or concepts, and even more when it comes to moving onto advanced math, because they never developed the realization that facts are used to solve problems--because, of course, they were never able to master the facts automatically in the first place.

In my view, these kids do *not* need long, extended explanations about how fractions work or why the ratio of sides in a 30-60-90 triangle is constant. In fact, that is exactly what they don't need. They need to stop having to think about these facts. If they can't internalize them as others do, then they will have to internalize them by rote.

Incidentally, I have given students math facts by rote and had a lot of success with it. I was working with kids in test prep, and only focused on problem solving in SAT or ACT environments. However, many students told me that they actually began to understand math and word problems for the first time, so they at least thought it was successful.

Cal,

When I say that we "celebrate failure", I don't mean that we are happy when kids fail, or that we tell them that it's ok. I mean that, instead of passing these kids along with a D because they are nice, we examine why they failed. We hold their own behaviors up to them to look at - for some students, it's a lack of effort, for others, it's a question of their classroom behavior, and for others, it's a problem with their numeracy or abstraction skills. And for many, these problems overlap. We confront their behaviors straight on, with both the student and the family when appropriate, and come up with a plan for change. Students fail our Algebra 1 classes not just because of a lack of numeracy and abstraction skills, but because they have so many other issues working against them. It's not uncommon for a teacher to bring a student up at assembly and recognize them for improving from a 30% to a 60%, and encouraging them to keep making positive choices. Yes, it's still failing, but that is a huge leap forward. And we have seen, again and again, students moving from 30% to 50% to passing, even to Bs and As, as they learn more about how to be a student. I think failure can be hard for students, but it's a lesson they need to learn. Of course, it is critical to provide them the right support to move them out of their long history of failing that they bring with them from middle school (we used to call it the "tunnel of Fs"). Our students are amazingly resilient, and an F doesn't deter most of them, once they are bought in to the idea of success. Students are actually quite perceptive about their own learning, and know when they didn't really earn their passing grade - and there's no way that charity grade makes them feel better than a hard-fought battle that didn't quite succeed.

So, I don't believe that their success in Algebra is purely a function of their ability to think abstractly.

As to whether abstraction can or can't be taught, I don't really have any evidence to say. My feeling is that it can be coached, and that there will always be some students who can think more abstractly than others, but that anyone can improve their abstraction abilities to some extent with proper training.

While I have seen that most of the time understanding wins out over blind memorization of algorithms, I do think there is a place and a time for that algorithm.

As a student I abhorred the algorithm; why should I memorize something if I don't even know why it works? Why should I even believe that it does work?

And yet I abhorred what I now know is called "constructivist" or "discovery" activities as well. I hated them because they forced me to work with others, forced me to pay attention and do some thinking. And I don't think my high school math teachers were very good at guiding the exploration.

When I got to college and my professors stood at the board, explaining each derivation step by step, I was in heaven. Now this is teaching, I thought. But the problem is, not all students learn like that. Not everyone is so interested in the WHY part that they will pay attention through a long derivation. Many people (and those people, I would argue, are the ones who tend to get things done) are just interested in the damn result - so they can use it to DO something.

Now that I'm a teacher of low-skilled (but improving) high school students, I can see the importance of making students think through reasoning for themselves. They don't always like it (in fact, they sometime hate it - they just want me to tell them "the answer"), but it is important for two reasons:

1) It develops logical reasoning skills, and gives students practice in their use. That means that when students are confronted with a problem in the future, they will be more likely to remember that they can think it through and find a reasonable answer. They will have more confidence.

2) It helps students remember and be able to apply the end result. And if that's not practical, what is?

I am primarily a chemistry teacher, so I'll give a chemistry example. The state standards specify that students should know that evaporation and melting are endothermic processes, while freezing and condensing are exothermic. I could just ask students to memorize these facts, and then to apply them. But they get so much more out of it if I have them do a guided exploration, with questions that build upon each other to reach the end result. This could be in the form of a lab or a worksheet or whatever, as long as they are asked to think about energy, temperature, states of matter and molecular motion, and then are asked to put it all together to figure out wheter evaporation is endothermic or exothermic. Then, as an application, they can explain why sweating is important for the human body. This exercise takes a lot longer than just having them memorize the answer, or even explaining the reasoning for answer to them and having them write it down. BUT now that they have gone through the reasoning themselves, they can call upon it when they get stuck trying to answer the same question in the future. AND they have practiced thinking through an issue logically.

But there are also times when I apologize to the class and tell them that we are just going to have to memorize something. Sometimes this is because the state standard is so trivial that it doesn't even merit explanation, but other times it's because I gauge that the rationale for the concept is something my students are ready for yet, or will not be able to master in the time allotted. For example, when I taught Le Chatelier's principle, we did go through the normal demos and scaffolding for equilibrium, but for Le Chatelier's principle itself I just had the students learn it and apply it. I did not ask them to understand its rationale because the rationale is very complicated and involves a lot of math and chemistry background that they students don't have and I don't have time to teach them. So they memorized the principle and just practiced applying it.

I used to believe that this kind of teaching was terrible; if you can't teach them why, don't teach it at all. But that's not the way it works. There are certain results that it's important for them to know about, and that they can learn to understand later. For exmple, most of us learn as kids that the earth orbits the sun and not the other way around, but only later in school (if at all) are we presented with the evidence that proves this fact.

As this is a blog for math teachers, I'll give a math example, as I teach Numeracy in addition to science. This year I observed that most students (9th graders) in my numeracy class were able to understand why we need common denominators when adding fractions but not when multiplying. They were able (and it was useful!) to use fraction circles and draw area models to figure out addition, subtraction and multiplication of fractions problems. But when it came to fraction division, students were able to understand what it meant for simple problems - for whole numbers divided by fractions or vice versa, or for fractions that went into other fractions a whole number of times, or a simple fractional number of times.

It was far more difficult to understand the connection between understanding what division means and the "flip and multiply" algorithm. This was undoubtedly related to the fact that it was the end of the year and we were out of time, as well as my own deficit in classroom management skills. But whatever the reason, it might have been better to teach the algorithm for fraction division, so that students would have adequate time to practice it.

So, to sum it up: guided exploration, constructivism and rationales if at all possible. But because this is an imperfect world, there are times when just teaching the algorithm is appropriate.

Wow. This is about the most illuminating discussion I've ever read on an edublog. Congratulations, Dan, on such a civil, thoughtful, interesting blog. I'll be back.

Thanks Rob, I hope you continue to find it useful.

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