I've had the same comment surface in several, unrelated conversations I've had with colleagues lately. All math teachers. In each case we were discussing some aspect of the curriculum and at one point they invariably say: "Y'know, you just have to memorize that."

Really?

Driving my son and a friend of his home yesterday we were talking. They started talking about tests they have coming up in various classes. They listed those classes where "you just have to memorize that stuff." Again, math was one of those classes.

So really, what do we absolutely have to memorize in math? I do not teach memorization, although I too have told my students that they have to commit certain things to memory. In each case I emphasize they should not memorize individual facts, rather, they should identify patterns and recall the patterns. I teach mnemonics. (Is that splitting hairs?) For example, how many patterns can you find in these two columns of numbers?

18

27

36

45

54

63

72

81

90

So really, what do we HAVE to memorize in math?

## 20 comments:

Darren..lets talk it through because I am preparing to have this very discussion with my faculty on Wednesday.

Straight wrote memorization may not be the key and I agree patterns are essential like 0.10, 10% and 10/100. I also believe for students to excel in high school math they need to have at their command basic math facts like those above and others. I believe that the ability to do mental math and to have quick recall of the basics is a real plus. My question then is how do we get there....

If we are building students ability to recall through minute math, mnemonic devices or rap.. in the end it is about memory but it is also about how we build and access the memory. So what are the best/ most effective ways?

Hi Darren - as an elementary teacher I hear suggestions from middle and high school math teachers about what we elementary teachers should do and cover:

1) "As a high school math teacher I would be thrilled if 9th graders appeared at my door that just had solid skills in addition, subtraction, multiplication, division of whole numbers... solid in place value to a billion, even just basic understanding of what fractions are ... simple knowledge of adding/subtracting fractions with like denominators and at least exposed to finding common denominators, have seen decimals and maybe even know the tenths place. Know basic "facts" in addition and multiplication and a few other basic math skills. If that happened I would be in heaven as a high school math teacher!" - My wife and I have several high school math teacher friends, and they pretty much agree on that description.

2) "As a middle school math teacher if they just know basic add, sub, mult, and general division we would take it from there!"

Occasionally I hear they want students to have more skills than that, but generally those the basic wants.

That is NOT however the reality of what we are expected to cover in elementary which is easily a twenty times more than that, so I would be interested in your take.

Brian

Tomorrow we begin our state assessments in math. In a collaborative planning meeting last week we discussed some startegies we've been teaching our students to help them do their best. My two colleagues have both been teaching their students to make multiplication tables, I have not. In our discussion I said that I would be interested to see some data on whether or not this makes a real difference. Here's why, if a student is actually capable of making a multiplication chart, do they really need to? Is there something about the actual act of creating it that would affect their scores?

I see this related in some ways to this discussion, but also straying a bit. I like how Barbara said, "access the memory." No matter how much I memorize if I can not recall when to open that bank up, the info stored in it is useless. While it is fantastic that I can create a multiplication chart, if I can not recall that it can directly help me with LCM or GCF, does it really do anything?

@Barbara You raise some good points. The example you gave (0.10, 10% and 10/100) is more about "multiple representations" than it is about patterning. MR is tough stuff. It's one reason why kids struggle with rational numbers.

E.G. 3/4 could describe a part of a whole (3 quarters of a cake) or it could describe a ratio (I have 3 smarties to your 4; in total we have 7).

The way I see it, there are two ways to think about memories in this context: Accessing memories and encoding memories. I think patterning is a much more powerful method of encoding than what is most often meant by memorizing i.e. the committing to memory of a number of disjointed facts using a sort of "brute force" method.

Memories built through patterning (story telling and metaphor can be powerful encoding devices) are easier to recall, IMHO. If you can encode a great many facts with a single pattern or story then you are more likely to be able to retrieve the memories when needed, particularly if the encoding method makes the memories sticky by having some sort of emotional component.

What I'm trying to get at in this post is: Do we really have to teach kids to memorize lots of stuff? Is learning math mostly about being able to memorize things? Or is there another, better, way of teaching and learning?

What would make the short list of things that can only be learned by the memorization of isolated facts? And would it be a "short" list?

@Brian I would tend to agree with what the high school teachers you've spoken to say. I'd probably throw in a little geometry too.

With that said, I'm not talking about identifying the basic skill set kids need for high school. I'm talking about how we teach kids math. Standing in front of the class when is it good pedagogy to say: "You just have to memorize this." Where by "memorize this" we mean to remember without understanding the contents' place in a network of connections/concepts.

Is there a better way? Can we teach better than that? Or are there certain things that can only be learned in this way? And if so, what do we mean by "learned?"

@Steve (Mr. Kimmi) I think you're much closer to what I'm trying to get at here. If kids don't understand how what they're learning fits in with everything else they've learned then how can they be expected to apply what they've "learned"? Your example is very much to the point.

That said, I think there can be value in having kids create multiplication tables. I'd be interested in having them do that then explain to the class "how" they did it. I would be most interested in hearing, from the kids, the different ways each kid went about doing it (the different patterns they used) and I think it would be a powerful learning experience for all of them to hear the similarities and differences in the approach taken by different students.

I can hear Eric Clapton playing ... "it's in the way that you use it ..." ;-)

We have been discussing just this issue at our school, as we consider numeracy across the curriculum. Maths teachers are concerned that teachers in other subjects simply give students rote routines without attempting to develop understanding. I have been questioning our moral high ground on this issue. It seems to me that we are happy enough as maths teachers to use rote routines in higher maths (whether or not we go through a process to explain why the routine works).

Memorization is an essential part of being an effective mathematician. We do out youngsters a great disservice if we suggest otherwise.

Anyone who relies on patterns to know the 9 times table (rather than simply recalling them from memory) is not operating at a high level mathematically.

I've run out of time here - I'll be back :-)

@Darren - It's funny that you should say, "the different ways each kid went about doing it (the different patterns they used)." Although I am not sure that there has been any sort of metacognitive follow-ups to the creation of the multiplication tables, the teachers used this thinking in the defense of or to justify their actions (it was hard to read their feelings about my statements). They said something to the effect that when making a times table the students more often than not considered it skip counting, and that for every column there is an identical row.

I think that there can be value in what they are doing, in fact at my previous school, which wasn't so test-centric, I taught my students to make a division table with decimals up to tenths. But even then I missed the metacognitive piece, except with the sevenths...we always nailed the sevenths.

So should memorizations be more of a Why and How game, an dless of a what?

After posting last night I was not able to return until this evening and quite a conversation is developing. We will be reading it in tomorrows curriculum meeting.

@Darren point taken about representations I guess it shows that I am an admin not a math teacher.Interesting point about the ratios too...something to talk about with the teachers and perhaps a good example of why understanding is so critical.

I get Mr. Jones point too. I think it speaks to what Mr Kimmi and I said about accessing information.

However it also seems true that "memorization" without understanding is useless so how do we find the balance between knowing and understanding?

IMHO we can only get there when we are teaching/learning in the area of application...I think we can do better and we need to do better because facts taught in isolation and never applied are soon forgotten.

Darren - I agree on the geometry (and I'm sure there would be a few other items to add to the list). It pains me to no end to watch as we try to shove math concepts at students before they're ready. I've watched it happen to my own daughters both of whom received straight A's in math all through elementary and middle school but they felt they were really poor at math. We don't build understanding, we just move on because we are behind the pacing guide the first week and to keep up we too often resort to teaching a step by step method quickly. Math is so perfect to explore and learn about and it seems that we ruin it for our kids and turn them into math phobes along the way.

Brian

@Robert (Mr. Jones) I think we're agreeing but saying it differently. I also think there are things students need to commit to memory, i.e. multiplication tables, exact values in the unit circle, etc.

Your point about maths teachers illegitimately taking the moral high ground is a good one. Unless teachers take a metacognitive approach to teaching content in their classes they end up teaching many things using rote memorization. You can teach rote memorization using patterning: In the unit circle we note all denominators are 2 and the numerators are all square roots from 1 to 4 out along the x- or y-axes. But I have never taught this until we've explored the two triangles included in every geometry set (one of which I say is a memorial to Hypassus), told the story of Hypassus, the Pythagoreans, and the origins of irrational numbers. There's also a discussion of similarity thrown in there so kids understand what works for triangles drawn in the unit circle applies to all triangles of any size.

As for replying on patterns to recall the 9 times table I do that myself. I can encode the entire table in my memory be remember two facts:

(1) The tens digit of the product is one less than the factor by which I multiply 9. e.g. 7x9 = 63

(2) The sum of the digits must be 9.

It's quick, it's simple, and to recall 10 different facts I need to remember only two as opposed to ten. Do you really think that's operating at a mathematically low level?

@Steve My answer to your closing question is yes. I think having kids draw up multiplication tables without also having them share/discuss/debate the way they drew them up is a waste of time. Or perhaps, better described as a lost opportunity.

@Barbara I don't think we balance knowing and understanding; I think they go hand in hand. I also don't believe that everything we learn must have a real world application.

For example, the is a mathematical object known as E8. It has 248 different dimensions (that hurts if you think about it too long; you and I are 3 dimensional creatures living in the 4th dimension: time) and it was "discovered" around 1890.

Recently, Garrett Lisi, a surf boarder physicist who lives in a minivan in Hawaii may have discovered the holy grail of physics: The Theory Of Everything. And it all turns on this purely mathematical object discovered over 100 years ago that has had no real world application until now, E8.

@Brian Amen brother. Amen. I work so hard at trying to heal the mathematically wounded kids that come through my class.

Sometimes I think math is used as a club rather than a key to unlock a door.

Thinking about those things in life which do require memorisation, such as Torah trope or PIN numbers, I realise that those are all artificial, quasi-random things with little or no pattern to them. The beauty of mathematics is that it does have gorgeous patterns in it, and it never ceases to amaze me that some of those patterns actually apply to the real world (NOT Torah trope).

Ideally, students would need to memorise very little because they would be able to use mnemonics or derive one expression from another. Unfortunately, the curriculum does tend to introduce things before the students are able to see the pattern. As an example, when teaching the behaviour of the trig functions in the upper two quadrants to the grade 10 pre-cal class, I brought in the entire circle, even though they wouldn't officially learn that until grade 11, simply because what they are supposed to learn makes no sense if you only have the upper half, and then you are reduced to rote memorisation.

BTW, I've had this conversation with English teachers regarding spelling. In fact, just about the EXACT same conversation.

Thanks for the food for thought!

@Dr. Eviatar I'd say memorizing using mnemonics is still memorizing. We can use mnemonics to memorize more efficiently but is there a difference between memorizing to be able to quickly recall facts and memorizing with understanding? The question I'm interested in exploring is: What sort of things MUST be memorized without understanding? When and under what circumstances would it be considered good pedagogy to stand in front of a class and say: "You'll have to memorize this." where by "memorize this" we mean "commit to memory these isolated facts."

I agree that it makes little (pedagogical) sense to teach the behaviour of trig function in only quadrants I and II. I might start there and ask the students to see if they can extend the pattern. This is a classic example, IMO, of a curriculum writer who views mathematics as computation rather than the science of patterns. Good teachers ignore that sort of nonsense and teach kids an a way that helps them make sense of what they are learning, helping to make explicit the underlying patterns of what is being taught.

And about the English teachers, do you think they have to put up with their discipline being thought of as consisting of nothing but grammar? English = grammar (or spelling; the mechanics per se) is analogous to mathematics = computation.

Can I bring about a slight shift in the conversation, because I believe we are all in agreeance that "to stand in front of a class and say: "You'll have to memorize this." where by "memorize this" we mean "commit to memory these isolated facts" is not good pedagogy.

What are some ways/strategies to create an environment where memorization is meaningful and connected with specific content. Often I have thought of creating a display that we build upon throughout the year, in which we have individual concepts spread around and we use string/yarn to connect things as we discover that they are indeed connected. Maybe post these concepts up along the cieling and build a sort of mathematical solar system.

But I'd also like to talk about how whether or not illuminating relevance increases students willingness to memorize.

Take your pick...or don't...please do however, this is helping me get through state tests.

@Darren,

I think it is an issue of curriculum integration. If we build things up carefully, we should be able to avoid the memorization. But I'm not sure it is possible.

Example: the volume of a sphere. We teach kids in grade 6 that the volume is 4/3 pi r cubed. I don't see any way for them to remember this except by memorization.

I had lots of fun with your AP Calculus class when they realised where the volume expressions come from. But how could I do this in grade 6?

Should we not teach volumes until they can do calculus? Or should we make up some kind of justification that would substitute for understanding?

Hadass (still posting as Dr. Eviatar, but you don't need to call me that, LOL).

@Steven I LOVE that yarn idea! What a fantastic way to make abstract concepts concrete.

You asked if "illuminating relevance increases students willingness to memorize."

I'd say illuminating relevance obviates the need for students to memorize. They engage with the content and it sticks.

I'm reading Made To Stick right now and your suggestion contains all the elements that go into making and idea stick: SUCESs:

Simplicity

Unexpectedness

Concrete

Emotional

Story

@Hadass Teachers don't get to build curriculum. (Teacher consultation vis a vis new curricula is largely a farce.) We teach what we're mandated to teach. Often that means we don't follow the curriculum when it exhibits poor instructional design. e.g. the grade 10 precalculus trigonometry you mentioned above. Regardless of the curriculum we're mandated to teach it behooves us to design learning experiences that are well designed. When needed, chuck the limits set by the curriculum. Never fear teaching kids "too much."

As for teaching solid geometry in grade 6 this can be easily done without using calculus.

You mention the example of the volume of a sphere. Teach it as an infinite number of cones all with the same height, the radius of the sphere. Then it depends on knowing two things:

(1) The area of a sphere, like a baseball, is four circles. In class cut the laces off a baseball, lay it out flat on the table as the kids gather around illustrate how it looks like 4 circles ... because it is. (Even a circle can be modeled as a parallelogram, and a parallelogram as a skewed rectangle.)

(2) The volume of any cone (or pyramid) is 1/3 the volume of the enclosing cylinder (or prism). This can be easily demonstrated in class using some hollow solids and coloured water.

Another benefit of doing this is it introduces the concept of a limit at a level that grade 6 kids can understand and prepares them for advanced study later; something that, with good teaching, is accessible to all kids.

All this talk of curricula, what types of textbooks do you use? Do they cycle?

We have a cycling text and it recieves a lot of criticism, mostly because it doesn't teach to the test. However, I think it does a pretty good job of building in this (somewhat) meaningful connection. While no text is perfect, ours does a good job on introducing, then reviewing at a time when there is a connection to something new.

For example, it may introduce adding fraction with like denominators, then LCM, next adding fractions with unlike denominators, then GCF, and finally lowest terms.

I'm wondering, as far as connections and memorizations go, do the text writers get it more right (not perfect, but better) than the curriculum writers (mostly based on the fact that they are more geared to what will be tested)?

As a college 7-12 math education major and being a tutor for math here on campus, I encounter this question a lot. sometimes I don't know how to answer them. I think there does exist some memorization because many times students I've encountered students that just want to know the formulas they have to use because when explained further as to why that equation works they get confused in the wording. Being a college student in math classes I can kind of see where the student is coming from but also where the teacher is coming from. As a student in my math courses many times I understand a concept well but it still requires some memorization on my part to know when and where to use it. We many times get introduced to a concept generally and then as we work through the semester we add to our knowledge which makes the concepts seem easier and not so abstract.

As a future teacher though I can see where we would not want to think that our students are just memorizing the information, because then we feel like they are not learning. Then again how do we know they are not learning maybe they have memorized it and know what its purpose is, and how it is to be used. Just because some memorization is involved does not mean necessarily that the student isn't learning because even though they have memorized information they are still building on the knowledge they already know and they will keep building on it throughout the time in school. I know this from personal experience in class because things I didn't understand a year ago I now understand to the point where I can now tutor others in the subject with complete confidence in what I'm doing.

I believe that Learning math is about 20% memorization and 80% application. I teach middle school Pre Algebra, and I surely appreciate when students come to me having already memorized their multiplication tables. The facts should be available for instant retrieval to be successful at higher levels of math. I understand that if a student "gets" the idea that multiplication is repeated addition, then he can eventually arrive at the correct product, however, it really hinders that students ability to be successful on application questions, and really slows them down. Memorization without understanding is useless. If you don't know when to use a particular formula or operation, then memorizing the formula serves very little purpose. In my seventh grade class I often teach new concepts using guided discovery, rather than simply putting the information out there. This allows for a deeper understanding of what is being taught. I do not require that students are able to derive formulas, but if they understand where the formulas come from they will have more success applying them (an memorizing them for that matter). For example, today in class my students learned how to use the area formula for trapezoids. If they memorize the formula, great, it will make their work easier, but as long as they understand the concept of area and have previously committed to memory the area formulas for rectangles and triangles, they will still successfully find the area. I truly belive that there is a place for memorization and that it benefits all students...but, there needs to be understanding behind it.

Thanks for chiming in Beth. You echo many of my own sentiments. ;-)

OK, I'm still not convinced. Yes, I know that as a teacher I'm still wet behind the ears, but my experience is that people will remember things if they need them. I'm seeing it right now with my son - so much agony trying to get him to memorise the times tables out of context. Now they are doing division (which he likes, funny kid) and he needs to know the multiplication tables and suddenly he does know them, because he is using them for something that makes sense to him.

Ditto for formulas - you use them, you'll remember them.

I do bow to your superior experience, but it doesn't make sense to me. Yes, of course mnemonics and so on are useful - but when you need something, not just to memorise because you are *going* to need it.

Hadass.

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