kitchen table math, the sequel: Why students have to memorize things

Monday, January 23, 2012

Why students have to memorize things

re: Larry Summers' claim that "in a world where the entire Library of Congress will soon be accessible on a mobile device..., factual mastery will become less and less important":

Larry Summers is wrong.

Factual mastery has not and will not become less important, for the simple reason that it is not possible to think about something stored on Google.

While you are thinking about something, that something has to be lodged inside working memory, not Google.

Biology does not work the way Larry Summers thinks it works.

Working memory

If I ask you to multiply 36 by 3 inside your head, working memory is what you use to do it.

Working memory (WM) does three things:
  1. Holds the problem -- "multiply 36 by 3" -- in consciousness 
  2. Retrieves the relevant knowledge from long-term memory (the times tables, in this case)
  3. Performs the calculation
Boiling it down, working memory is:
  1. a form of storage
  2. a search engine 
  3. a "computer" or thinker
"Critical thinking" is accomplished by working memory.

3 to 5

The fact that we can think only about things stored inside working memory leads directly to the need for "factual mastery."

Factual mastery—knowledge stored inside long-term memory—is essential because although long-term memory is vast, working memory is tiny:
...cognitive tasks can be completed only with sufficient ability to hold information as it is processed. The ability to repeat information [you have just heard or read] depends on task [difficulty]... but can be distinguished from a more constant, underlying mechanism: a central memory store limited to 3 to 5 meaningful items in young adults.

The Magical Mystery Four: How Is Working Memory Capacity Limited, and Why? by Nelson Cowan
Working memory can hold three to five items at once. That's it. That's the limit.

Three to five.

I hit this limit all the time trying to write about new topics. The basal ganglia, for instance. For well over a year, I have been endlessly working and re-working a project on the basal ganglia, a subject I knew essentially nothing about going in. Where the basal ganglia were concerned, my long-term memory was a blank slate.

The upshot: I was not able to write about the basal ganglia until I actually learned about the basal ganglia: learned as in committed the material to memory. It didn't matter how many times I looked up basal ganglia on the internet. I looked up the basal ganglia on the internet a lot, as a matter of fact; then I forgot whatever it was I had looked up while I was looking up something else to do with the basal ganglia, after which I'd have to go back and re-look up the first thing all over again.

Try it if you don't believe me.

Here are some terms related to the basal ganglia:

Dorsal striatum
Ventral striatum
Nucleus accumbens
Ventral tegmental area
Orbital frontal cortex
Two pathways
Probabilistic learning
Associative learning
Statistical learning
Orbitofrontal cortex
Cortico-striatal circuit

Now supposing I handed you a laptop and asked you to look up each term on Wikipedia, then write a coherent, reasoned 5-paragraph essay on the basal ganglia: what it is and what it does. Just a quick summary organized into 5 coherent paragraphs.

You couldn't do it.

You couldn't do it because every time you wrote about the ventral striatum, the dorsal striatum, and the orbitofrontal cortex, you would forget the VTA and the putamen—and you would forget the VTA and the putamen because your working memory will hold only 3 to 5 things at once. Something has to go.

That's what happened to me when I took the SAT with a calculator I didn't know how to use. Each time I swapped the steps for using the calculator into working memory, my brain swapped the information for the problem I was doing back out of working memory. Then, when I tried to cram the information for the problem back into working memory, the calculator steps got squeezed out again.

I could remember the problem, or I could remember the calculator, but I couldn't remember both at the same time. Too much information, literally.

My calculator fiasco illustrates the reason you need to practice until you learn content and skills to the point of 'automaticity.' (Automaticity is another basal ganglia term, by the way. The basal ganglia are the part of the brain that underpins automaticity.) Once you've learned something so well you don't have to think about it, you free up space in working memory to hold other things.

Thus if you know the times tables "by heart," you don't need to pull "3x6=18" into working memory. Working memory can locate "3x6=18" inside long-term memory and use it without displacing "36x3."

Knowledge stored inside the brain is different from knowledge stored outside the brain

Experts always possess factual mastery of their fields. Always.

The reason experts always possess factual mastery of their fields is that knowledge stored in long-term memory is different from knowledge stored on Google.

Knowledge stored in long-term memory is (or becomes) biologically connected, or "chunked." Thus to an expert on the basal ganglia, ten facts about the basal ganglia are just one or two big facts about the basal ganglia.

Chunking is the magic, because working memory doesn't care about chunk size. Working memory can hold 3 to 5 small and simple items or 3 to 5 large and complex items. Either will do. Chunking gets around the limits on working memory.

Dan Willingham's demonstration of working memory

For a demonstration of the chunking principle, read the list below, then look away and try to remember what you've read:


How many letters did you recall?

To find out how many letters you would have recalled via prior chunking inside long-term memory, see Daniel Willingham's explanation in "How Knowledge Helps" (American Educator | Spring 2006).

(The answer is all of them.)

You can't Google knowledge chunks

Knowledge chunks can be created only inside the brain, via learning. You can't Google someone else's complex knowledge chunks and swap them into your own working memory. It doesn't work that way. Your own brain has to do the work of chunking, and your brain does that work through the process of learning, bit by bit and step by step.

Which means that the process of storing content in long-term memory is not a simple matter of "memorizing facts" so you can "regurgitate" them later.

Over time, memorization creates the complex knowledge chunks that allow knowledgeable people to engage in complex thought.

Experts think better than novices because experts have factual mastery

To a gratifying degree, I can now think about nearly all 19 items on the basal ganglia list at the same time. I'm still struggling with "putamen" and "ventral tegmental area," but the other 17 are stored in memory: my memory, not Google's. So, for me, those 17 items are no longer 17 separate items, but closer to 2 or 3. When I think about 1 item on the list, I'm thinking about the others.

I reached this point by committing these terms and concepts to memory. As the terms entered my long-term memory, they became biologically connected and chunked. Now that I can think about them at the same time, which means I can write about them, too.

What makes experts expert, to a large degree, is factual mastery of their fields. Factual mastery allows experts to think deeply and well because the content they are thinking about has been biologically connected and chunked inside their brains, and there is no obvious limit to the amount of chunked content working memory can manage so long as knowledge has been chunked into no more than 3 to 5 separate entities.

Factual mastery is required for complex thought.

Which brings me back to Larry Summers.

If our schools are going to ask students to 'think' about material they haven't learned, students are going to be thinking about 3 to 5 small, not-well-elaborated items at a time. Period. Their thinking will be superficial, and the conclusions they reach will be superficial, too.

Which is exactly what we see in Larry Summers' op-ed about education, a field in which he is neither expert nor learned.

Superior Memory of Experts and Long-Term Working Memory (LTWM)
Extremely fast learning & extended working memory
The Number and Quality of Representations in Working Memory by Weiwei Zhang and Steven J. Luck
How Knowledge Helps by Daniel T. Willingham American Educator Spring 2006



Catherine Johnson said...

this topic is quite challenging to write about, and I haven't quite nailed it

so I'm adding an addendum

I want to highlight the fact that knowledge offers TWO ways around the limitation on working memory:

1. Well-ingrained knowledge allows you to free up working memory. I don't understand how this works; not sure if anyone does. But somehow, very well learned knowledge can be accessed and used even though it's not conscious (or doesn't seem to be conscious). 'Automated' knowledge seems to run without having to enter working memory.

(take that with a grain of salt)

2. The more expert you become in a subject, the more highly elaborated the 'chunks' of content you can bring into working memory. This aspect seems better nailed-down to me, and a bit more understandable. A real expert can hold a huge amount of material in consciousness - in working memory - at the same time because it's all the 'same thing.' He's got the same 3 to 5-item limit, but the items are MUCH MORE DETAILED.

Grace said...

Even the co-founder of Wikipedia has warned us that the only way to begin to know something is to have memorized it.

Wikipedia co-founder says we need to memorize things, not just ‘Google it’

Debbie Stier said...

I'm forwarding this on to Ethan, who, once again, underperformed on his mid-terms.

He did manage to position the grades well though, via text message from school; it was only when he arrived home, and told me what the grades actually were, that I told him that he'd over sold and under delivered.

He clarified that it was a new wording bias skill he'd learned in his statistics class.

Hainish said...

On point #1, yes: having information in long-term memory (i.e., memorized!) something frees up working memory because (I'm inferring) only the new stuff that you're thinking about is occupying the 3-5 slots in working memory. When you were learning about basal ganglia, you didn't have to hold concepts like "brain" and "central nervous system" in your working memory.

Have you taken a cognitive psychology course or read a cog sci textbook?

SteveH said...

There is a lot of wiggle room in his list, but it seems to be taken from the standard K-12 educator's playbook.

Here is his list decoded.

1. Mere facts.
2. Group work
3. Technology
4. Active, not passive learning
5. Global, multicultural
6. Data Analysis

I was looking for a little better insight tempered by his tenure at Harvard, but apparently he never talked to the parents of the students who got accepted there.

Technology is great. It offers lots of opportunites. However, you have to look at the details. Is the technology used to increase expectations or is it used to avoid work?

I was in college when calculators first came out. They allowed us to tackle complex algorithms that were not possible with the slide rule. Some calculation homework assignments took 40 pages. What do we see in K-6? Calculators are being used to reduce the importance of learning the times table and other basic skills. It's being used as an avoidance tool.

I don't have to be able to do long division quickly by hand and I don't have to memorize the list of presidents, but ... where is the cutoff?

My son loves to memorize things. He just finished memorizing all countries and their capitals. Of course that's not necessary, but what geography do you need to know off the top of your head? How can you look up something without a place to start? Once you Google the stuff, what do you do with it?

Nobody argues with the need to know some things, but what is that point? Once again, it's argue with generalities, but they get to decide on the details.

Luke Holzmann said...

I balked a bit at "memorize" because what you describe here isn't exactly memorization (at least, not the way I think about memorization). There is a big difference between learning something and memorizing it.

Memorization (working definition off the top of my head): The ability to recall specific details on demand.

Learning (again, off the top): The ability to apply understood material to a context different from its original presentation.

Example: I was in Awana growing up. We memorized a ton of Bible passages. I could rattle off the words and did quite well at memorizing. But all that didn't help me think or write or talk better about Scripture. It wasn't until my leaders prodded me to not just recite but explain the meaning that I started to learn and apply the passages.

Granted, I agree with you that I had to learn the passages before I could do that. So memorization is important at the start. But I postulate that we tend to recall things we encounter on a regular basis and have need to remember. I don't work at memorizing how to drive somewhere. I do it a few times and if it's a place I need to go to again, my brain will generally latch onto it.

In fact, memorization only goes so far. If we don't use a particular bit of data, we tend to lose it. For example, while I mastered mathematics well enough to get a near perfect score on the math portion of the SAT, much of that is no longer in my working memory. I could, probably, quickly relearn it... but I would have to brush up again. Still, the ideas behind math stick with me, even if the specifics have long faded in disuse.

Sorry for rambling through this. I'll try to summarize: Memorization is a valuable tool for learning, but it is not learning or even very practical in and of itself. Learning (including memorization), on the other hand, happens naturally when material is presented in a way that proves useful and/or practical to the current situation.


Glen said...

Google no more reduces the need to memorize facts than a hardware superstore next door reduces the need to install doorknobs and faucets in your home.

Google is an information superstore. A superstore makes building easier. It doesn't make it less necessary.

SteveH said...

"So memorization is important at the start."

What is the difference between memorizing things and remembering things? What, exactly, does rote memorization mean? If someone tells you their name and you remember it, is that a bad skill? How about for a group of 4 people? Is memorization better if it's easier? How about HOMES for the Great Lakes? Nobody argues against remembering things. The more things you remember, the better.

Modern educational thought puts facts into a secondary roll. They want to make learning easier. Facts are learned in a thematic context - they are rarely learned explicitly. The assumption is that they will be better remembered. The problem is that they never test that assumption. It's like doing problem solving in math and assuming that mastery of the basics will automatically get done.

What if I had an easy technique to remember all of the presidents. Should a school ignore it because it's just rote learning or mere facts? Nobody argues for only learning facts. It's a strawman to claim that that's what tradiditonal learning is/was about.

For many things, like history, I don't see things unless I have some related dates and facts surrounding them. I tie understanding to facts, not the other way around. If I don't have enough facts, I don't see things. My son and I call this the "hiatus" effect.

This is the idea that things are invisible until some key thing (fact) triggers related information in the brain. In the case of the word hiatus, once he learned the word, he saw it everywhere. This has happened to him for lots of things. It's almost as if the word never existed before. He claimed he never saw it before. Now, he sees it everywhere. Obviously, reading the word in a book didn't translate into any factual knowledge or even remembering that he had seen it.

I'm reading some books on the war of 1812, and I see the effect happen all of the time. It's not becasue I know the social and economic forces that led to the war. It's because I have many more facts to tie everything together. My understanding rests on my framework of facts. The problem with rote learning is that some of the facts disappear if you don't reinforce them with other connections. It doesn't mean that you turn the learning around and deprecate facts as "mere".

I claim that knowledge and understanding are tied to facts, not the other way around. They may reinforce each other, but there has to be some explicit memorization going on. Thematic learning won't get the job done.

In first grade, my son had a thematic unit on Sands From Around the World to help with learning about geography and a variety of other things. One of the containers contained sand from Kuwait. He had to show the student teacher where Kuwait was on the map. His regular first grade teacher said that his knowledge was "superficial". I call it foundational.

One could argue that with computer technology, you don't need to memorize as much. With a library nearby, I don't need to memorize as much. Now we're talking about being lazy or not. I don't think anyone is memorizing the difference between their local library and the Library of Congress.

Online information is wonderful, but I don't see how it relates to having to memorize less stuff. Specifically, what facts (currently taught in school) aren't needed because the Library of Congress will be online?

Cranberry said...

I've started to notice that discussions of current events are framed through references to history.

If you google "missouri compromise," "Great Depression," "Gettysburg Address," "Maginot Line," you will see discussion of other matters which take an acquaintance with history for granted. You won't understand the full argument if you don't have a grasp of history.

Limit the search to news, and it's fascinating. Sports writers seem to throw out historical terms without explanation. The Cowboys defense makes the Maginot Line look pretty good. They give up more ground than Neville Chamberlain. Need I go on? Defense still wins in college football, as horrible offense-minded rules are not yet as pervasive as in the NFL.

One may be able to Google for information, but you need to know what you're looking for. You need to know what you don't know--and that calls for individual knowledge, not collaborative conversations. In my opinion, references to history are very economical modes of speech. There's a lot of information locked up in comparisons to history. The world moves too quickly to wait for the ignorant to be brought up to speed on Dred Scott.

Anonymous said...

"What, exactly, does rote memorization mean?"

I can maybe provide an answer (maybe because I'm not on *that* team). I think that some of them want to say "memorization without context." Maybe.

There is an interesting 1943 article in the New York Times bemoaning the poor grasp of history that college students possessed at the time (the article made it clear that they expected that the students would have *learned* the history in high school, but clearly retained little of it).

One example that they give of something that the students did not know (only 2% got the right answer) was the price before the passage of the Homestead Act of the "minimum price for an acre of Federal public lands sold at auction." The authors of the article seemed dismayed that this information had not stuck with the students for several years after they had learned it.

I can't get all that worked up over this horrible knowledge gap.

Most of the kids also didn't know the original 13 colonies (and the mistakes were pretty bad ... interestingly, it is clear that the test was free response, not multiple guess) and I'd consider this a much more serious gap.

But at least in 1943 both were considered a problem worth reporting. This suggests that the education folks have some history (pun intended) of teaching history as a pile of context-less facts. Getting away from this seems like a good idea.

But they overshot (partially because some/most? of them want to get away from learning *ANY* facts). What/how the kids were being taught in 1943 was clearly not working, though.

-Mark Roulo

SteveH said...

"want to get away from learning *ANY* facts.."

Their main goal is to reverse the pedagogical direction. They don't want to define that fact line. Instead of trusting the spiral, they trust the theme. My son's schools have been all about reading, reading, reading, and they have not been too particular about what the kids read. My son reads a lot but I'm surprised at how many words he doesn't know. They want implicit education rather than explicit education. It's a way of sounding important while setting lower expectations. All kids can learn, but they are not going to get much help from the guides-on-the-side.

Michael Weiss said...

Here, I think, is an example of "rote memorization." This is a true story, honest. I am currently tutoring a 5th grader in math. The first day we sat down together, I drew a rectangle, marked the width 8 and the height 6, drew horizontal and vertical gridlines inside it to partition it into a grid, and asked him: How many little square are in the grid?

He couldn't answer, not without counting them one at a time. When I told him not to count them, he was just stuck. 100? 20? 16? He had literally no idea what to do.

Then I asked him what the area of the rectangle was. Without any hesitation he said "48". I asked him how he knew that, and he said, "Don't you multiply them?"

So here is a student who has memorized that Area = Length x Width, and has also memorized that 8 x 6 = 48. But he doesn't understand either of those two facts. It's pure traxoline.

This is what folks are talking about when they rail against "rote memorization" and say that it's understanding, not facts, that matters.

Of course the alternative position -- that as long as you can look up the formula for area, and calculate 8 x 6 on a calculator, you are fine because you "understand" what to do and can just access the facts on your remote gadget -- is just as ridiculous. But that's the problem with our discourse about education: It always gets expressed as either/or, rather than both/and.

Jen said...

I'd argue that your student's problem is the easier to solve though. He's got two well ingrained bits of useful knowledge. He knows them cold.

Now, yes, should he also know that third bit? Surely. But because he knows the multiplying and the area formula, adding ON a third bit of info is amazingly easier! (And then later adding on that you can likely cut many straight-edged figures into rectangles and triangles to find their area and then add on...)

A set of questions worded various different ways, a couple of pictures or different, novel ways of asking the same question, some questions working backward (this square has sides of __, how many of these squares would fit into the shape above?)

That's how learning works -- by adding on, and on, and on. Is he flexible enough to identify area problems in novel situations yet, no. But is he on his way due to knowing those other pieces? Yes, and if he didn't have those other two pieces, he would be nowhere near close.

MagisterGreen said...

"So here is a student who has memorized that Area = Length x Width, and has also memorized that 8 x 6 = 48. But he doesn't understand either of those two facts."

I'd say instead: But he doesn't understand how to apply either of those two facts in a novel and unfamiliar situation.

Or perhaps: But he doesn't understand how either of those two facts broadly relate to each other.

This isn't so much a critique of what is called rote memorization as much as it is a critique of making proper use of that knowledge.

One caveat: As a 5th grader he's what...12? 11? Developmentally speaking, then, he may not be able to abstractly think about those two facts he knows to apply his knowledge in the situation. So we may not be seeing, in this example, an example of memorization vs. understanding as much as a youngster who isn't quite yet capable of manipulating facts in an abstract manner.

lgm said...

Well, I'll give you a rote memorize example. On a blank sheet of paper, list the states, their capitals, the post office abbreviation for the state, and their major bodies of water. This is for American History to Civil War via the SUNY Early College in the High School Program. This quiz is 10% of the college grade. I crossed SUNY Albany off my list as that is not what I want for a college level history course.

Anonymous said...

lgm, that's not even a History question, it's a Geography one (and not a terrible one in my opinion; let's assume that there are some additional questions that use that information to analyze issues and draw conclusions). As a History question it's a real dog; the states entered the union at varying points before and after the Civil War, and many have had more than one capital during the time that they have been colonies, territories, and states.

Anonymous said...

"that's not even a History question, it's a Geography one (and not a terrible one in my opinion; let's assume that there are some additional questions that use that information to analyze issues and draw conclusions)."

Let's *NOT* assume that. It might be true, but I'm not willing to assume that without evidence.

But I also don't think it is a terribly good question. I think it is reasonable that a child can find states on a map (both in the "where is California" form and in the "what State is this" form). But I'm not super impressed with the ability to *list* the states from memory. Less useful seems to be to know the capitals ... I'd much rather that the kids know the *major* cities ... which are often not the capitals.

But my real complaint with this sort of question is that it often is the sort of information that the kids learn for the test and then forget. In some sense this isn't a problem with the test or even the class per-se, but is a problem with the entire framework. If we want the kids to remember this they need to review it every year for a number of years. If we don't care if they remember it, then I'm not very clear on the point of the test.

-Mark Roulo

lgm said...

I agree with Mark. There were three other fill in categories on this particular test, only one of which related to the history that they were supposed to be studying.

The time spent memorizing detracted from the time needed to figure out how to write acceptable papers, and to read and analyze the class material. I would much rather have had my kid assigned to spend his time with the primary source documents instead of making maps and flipping thru flashcards to memorize data tables. They were not asked to do anything with this material other than spit back on the test. The other tests did have timelines to fill out, which I don't consider to be rote as they were supposed to have studied, discussed and come to know the material.

SteveH said...

"So here is a student who has memorized that Area = Length x Width, and has also memorized that 8 x 6 = 48. But he doesn't understand either of those two facts."

Do you think that schools teach facts and stop there? Look at any proper math textbook and you will see problem sets that approach problems from many different directions. They work on building flexible knowledge and understanding up from mastery of the basics. That some kids don't get there is not resolved by reversing direction and starting with some sort of understanding first.

Having taught college algebra, I saw lots of examples of what I would call a rote application of rules. That's indicates a poor student. They probably didn't do the homework. They are pattern matching and trying to plug and chug. It doesn't work and they will get failing grades. It's some sort of myth that traditional math somehow allows you to get passing grades with rote knowledge.

However, if students come at the problem from the understanding side, what, exactly, do they understand? How do you ensure mastery of the basics? If you don't do the problem sets with all of their interesting variations (drill and kill), what do you really understand? You will know only concepts that melt away as soon as the math becomes more abstract.

When you study place value for numbers, what do you understand, that the '4' in 1640 means 40, or do you understand this from an algebraic understanding? Do you know scientific notation? How about different bases? Do the students understand it enough to create their own numbering system in any base, using their own made-up symbols for numbers? If kids have a pie-chart understanding of fractions, what happens when they get to rational expressions?

Rote memorization is used as an excuse to approach learning from the top down with less emphasis on mastery of the basics. You may get away with this in history, but not math. Understanding in math is a different animal.

SteveH said...

"On a blank sheet of paper, list the states, their capitals, the post office abbreviation for the state, and their major bodies of water."

Since this is geography and not history, I would advocate more memorization than other subjects. The cutoff point is subjective, but I would expect that students should be able to fill in a blank map of states with their names and their capitals (not where they are located). I wouldn't ask for their major bodies of water, but I would have them fill in (on the map) some of the major bodies of water of the country.

If you put Arkansas up near Nebraska how much understanding will you have when you read about these places. If you read about westward expansion but don't have some factual framework of geography, what will you understand? Facts are not built on top of understanding. If you don't approach facts in a sytematic fashion, you won't have a proper framework for understanding.

The process is iterative and can be approached better than the rote memorization of a lot of facts first, but this won't happen when educators view facts as superficial and mere. Nobody argues that remembering is bad. The more you remember, the better. The memorization of a lot of material without other supporting facts or information (or use), becomes a waste of time. The facts are not. However, one cannot depend on osmosis to learn all of the facts you need.

SteveH said...

By the time you get to college, you should have most of the states and capitals memorized. Students have had 12+ years to learn these facts by osmosis or any other method. Students are not sitting there thinking that they have to memorize 200 pieces of new information. After memorizing their missing gaps for the test, they might forget it all, but I doubt it. It will come back easier the next time. If they are not expected to memorize the information, then how will it get done? Whether this is good information to remember or not is another subject. Educators who claim that all facts are "mere" will do a poor job ensuring any level of remembering facts.

momof4 said...

By college? How about 5th or 6th grade? My kids knew them by 4th, at least. Working on that sort of basic knowledge was a routine part of car trips, along with math facts, spelling and American folk and patriotic songs. Of course, that was before the Walkman and Game Boy, let alone all the current electronic babysitters. We actually conversed (!) in the car. The kids even enjoyed it, as we parents did.

Michael Weiss said...

"So here is a student who has memorized that Area = Length x Width, and has also memorized that 8 x 6 = 48. But he doesn't understand either of those two facts."

I'd say instead: But he doesn't understand how to apply either of those two facts in a novel and unfamiliar situation.

Or perhaps: But he doesn't understand how either of those two facts broadly relate to each other.

Actually I'm going to stick with: he doesn't understand those facts AT ALL. They mean nothing to him. Having worked with this student for some time now, I can attest that when I got to him, he did not know that (for example) six bags, each with 8 apples in them, makes a total of 48 apples (he would usually answer 14, because "total" means "add"). When I asked him to give me an example of when multiplication would be used, he would say "If you have to find out what one number is times another."

Look, let's do an experiment. I'm going to define a new operation for the numbers 1-4 -- let's denote it with the @ symbol.

Please memorize the following table of values:

1 @ 1 = 4
1 @ 2 = 3
1 @ 3 = 1
1 @ 4 = 3
2 @ 2 = 4
2 @ 3 = 4
2 @ 4 = 1
3 @ 3 = 4
3 @ 4 = 1
4 @ 4 = 2

All other combinations are handled by stipulating that the operation is commutative.

Now: Could you commit that table to memory? Sure, maybe flashcards would help. Would you understand what @ means? No, I don't think so, because the values are (quite literally) randomly-generated. There is no meaning attached to the symbol at all.

That's what "6 x 8 = 48" means to this 5th grader. It's a thing he can recite by heart. It has no meaning at all. "Times" is just a word.

He is an extreme case, but there are more kids like him than you think. And I cannot for the life of me see how being able to recite "2 @ 4 = 1" in any way prepares somebody for later understanding.

Michael Weiss said...

My five kids are all home-educated, and every single one of them by age 7 (well, the ones aged 7 and up, at least) could answer the question "How much is eight 14s?" using purely oral and mental methods before they knew by rote that 8 x 4 was 32, and before they knew any written algorithms.

I would ask questions like this while we were on walks, and without any prompting or explicit teaching they would so some variation of the following: "Well, eight tens is 80, and eight fours is... well, four fours is 16, so eight fours is twice as much, which is... well, two fifteens makes 30, so it makes 32. And 80 and 32 is (ninety, one hundred, one ten...) One hundred and 12."

Horribly inefficient, and almost guaranteed that you'll make a mistake somewhere, because working memory only gets you so far. There is a reason why written algorithms are superior to mental math. And it took too long, and by the time they were eight they would do this on paper using standard methods. So I'm not advocating for invented algorithms as better than or a substitute for standard ones. My point is that the understanding came first, and the facts & algorithms later.

FedUpMom said...

I'm well into middle age, and I've never learned the state capitals. I can't say it's held me back any.

SteveH said...

"By college? How about 5th or 6th grade?"

I was trying to be nice. One of my big introductions to educational thought happened when I mentioned to my son's first grade teacher that he loved geography - that he could find any country in the world. She said: "Yes, he has a lot of superficial knowledge".

This is not a war against rote memorization. It's a war against facts and remembering.

Glen said...

And I cannot for the life of me see how being able to recite "2 @ 4 = 1" in any way prepares somebody for later understanding.

If your '@' operator is a variable standing in for something necessary but not sufficient, such as "times table" multiplication, then I'll tell you how it prepares them: by giving them a necessary prerequisite.

If you have a school class 'C' that has, as prerequisites, classes 'A' and 'B', and you've taken 'A' but not 'B', you aren't yet prepared to take class 'C'. This does not mean that your class 'A' did not prepare you in any way for class 'C'. Of course it did. Just because class 'A' was insufficient, it does not follow that it was unnecessary. It was a prerequisite.

To use multiplication in the real world, you have to understand what multiplication does for you. You also have to have your multiplication tables memorized so that, ironically, explanations of USES of multiplication can be "understood" directly rather than taken on faith as "facts to memorize."

Of course a proper sequencing for multiplication builds up using applications where you count the answer enough times that the usefulness of just remembering certain common combinations of n x m becomes obvious. "So you don't have to count every time. You can count anytime you want, but you'll get pretty bored counting the same thing over and over again, so it's easier if you remember some of the common answers...."

I assume you use some form of this iterative, tie it all together, approach, where the multiplication table is presented more sensibly, but it's still memorized.

Someone who has memorized the table without knowing how to use it has done SOME useful preparation, just not enough, and the same can be said for someone who has the "basic idea" of what multiplication is without having memorized the multiplication table: good start, but not enough.

And on another topic: I loved your "traxoline" example, but I have a feeling it meant something different to me than to some ed school professors. I kept expecting a closing paragraph promising that if I didn't understand it all right away not to worry, because it would all be repeated verbatim the following year--and again the year after. Why carefully sequence a linear course of taking what you already have mastered, extending it, mastering the extension, further extending it, and so on, when you can just "expose" them to tons of stuff, have them discuss it amongst themselves, and move on, promising to do the same thing year after year until: UNDERSTANDING! Just trust the spiral!"

SteveH said...

"My point is that the understanding came first, and the facts & algorithms later."

What understanding? What facts? Obviously, the facts of numbers came first. And the facts of counting. And the facts of adds and subtracts to ten. People don't like rote memorization because (perhaps) too many facts are learned first before they are tied to some other facts, knowledge, or understanding. Nobody argues with remembering. The problem is that the technique used might not work - the facts might not last.

It's quite another thing to get pedagogically weird about facts as if they are mere or superficial. That's what schools are doing. Tradidtional education doesn't mean lots of rote facts and little understanding. However, many educators want to use that strawman to change education completely around to talk about understanding and not mere facts. This is actually worse. Students still can't do the problems and their understandings are not developed enough to be successful when the math becomes more abstract.

By the time kids get ready for multiplication, most have developed (or been exposed to) some level of understanding about multiplication. Five times seven is five groups of seven or seven groups of five. They know something about place value (not just from some self-developed understanding) and that 14 is 10 plus four. By the time they get to to officially learning the times table, they already have some level of understanding or else the school or the child is having real problems. Learning the times table facts does not just appear out of the blue.

At some point, however, certain skills have to be mastered. You can talk all day, and kids can use their limited understandings of something to kind of figure things out, but how do you get to the next level of understanding? It's not from the top-down. You don't just study and teach more understanding. The proof of understanding is in the solution of a variety of problems that cover all different angles.

Modern educational thought thinks of this as solving open ended problems that might not have one answer. This is only a very small part of understanding. I'm talking about detailed and abstract understanding about how math works. The fact that many kids try to tackle problems in a rote or plug-and-chug fashion is not a problem of missing some sort of high level understanding. It's because they never mastered understanding from the bottom up.

If a student can't do two-digit multiplication reliably (using any method), how do you fix that with understanding? You could try partial products (That's what Everyday Math uses.) but talk is just talk. The proof is in doing the math. Mastery of algorithms is not just about speed.

This might seem wrong with the times table (which many learn by rote even though they could come up with an answer using understanding), but many educators' ideas fall apart when you get to the abstract nature of algebra. How do you solve an equation by rote? How does a simple pie chart understanding of fractions evolve into manipulating rational expressions?

When you are studying the D=RT governing equation and all of the problem variations you can get, understanding first can only be superficial. I could talk about how you can't average speeds, but students will only understand that by doing lots of problem variations. Also, this does not require in-class group learning. I have had lots of light bulb discoveries with standard homework sets.

Complaints about rote memorization are used to justify all sorts of changes to education. The times table is a special case. Speed is so important that one cannot leave it to understanding. If some kids end up with no understanding of multiplication, that's not because of some fundamental pedagogical failure in traditional education. That's just bad teaching.

Resonance said...

I am joining this discussion late so I hope my contribution is noticed. I propose a question in regards to memorizing versus understanding. Lets say you give two students a calculus problem. Student A has memorized all of the rules of differentiation and student B has not. Student A begins to go through each rule one by one until he/she gets the derivative. Student B looks at the problem grabs a reference sheet for differentiation and scans for the rule that best fits the problem and finds the derivative. Both students have the correct answer in roughly the same amount of time. Which student "understands" differentiation better? Which student would be considered more intelligent? Another way of asking this is which is better memorized knowledge, or the intelligence to know how to find what you need to solve the problem?

Jen said...

"grabs a reference sheet for differentiation and scans for the rule that best fits the problem and finds the derivative."

1) What happens without the reference sheet?
2) How does the student know which rule "best fits" the problem?

The way to know which fits best is to do what Student A is doing, over and over. Soon, student A will begin to notice the things about problems that make them better fits for one solution than another. S/he won't have to go through a list, but will have that sense of fit.

Student B won't get the correct answer at all without a reference sheet, certainly not in "the same amount of time." Student A is on the way to being able to find a best fit all on his or her own, through the most boring of technique, practice, practice, practice.