kitchen table math, the sequel: My dean teaches math to fifth graders

Friday, July 20, 2007

My dean teaches math to fifth graders

I've been attending a mathematics lab class taught by my dean. I've posted about it here, here, and here. Thought folks might be interested
Posted by at

21 comments:

Barry Garelick said...

I assume your dean is Deborah Ball. Although you are careful to say that the problem with the 8's is not a brainteaser or a puzzle, I disagree. On that topic, Hung-Hsi Wu, the mathematician at UC Berkeley has this to say:

"A judicious use of mathematical puzzles has its place in a
mathematics curriculum as a tool for training mental agility."

He goes on and does offer the following warning that frequent use of such puzzles "without any preamble can only reinforce the popular (and unfortunate) misconception that mathematics is nothing but a bag of cute tricks. It would be far better if each puzzle is prefaced by a disclaimer to the effect that “This is a test of your ingenuity”."

Although Deborah is not making such puzzles part of exams, I think his warning about doing so is worth reading as well:

"Incidentally, I would hesitate to recommend making puzzles part of an examination. ... An examination should test only whether a student has learned well, not whether she is inspired at the particular moment of exam-taking. To most of us, solving a puzzle does require inspiration."

His note about inspiration is worth noting. In contrast there are problems that require flexibility in thinking in that one needs to apply prior knowledge to a new application. This requires mastery of content knowledge and doesn't rely quite as much on the intangible. This is a problem from the 5th grade Singapore Math textbook:

"Mr. Anderson gave 2/5 of his money to his wife and spent 1/2 of the remainder. If he had $300 left, how much money did he have at first?"

This can be solved without algebra. Would this be a good problem to give to the students in Ms. Ball's math lab? If they have mastered the appropriate content knowledge, I would say yes.

July 20, 2007 at 6:25 AM
SteveH said...

"My dean is teaching math to 27 rising fifth graders and using her work with them as a laboratory for learning about teaching and learning."

She might learn something about teaching "rising fifth graders". Or, maybe not.


"A second problem is the eights problem. Students were asked to make a mathematical expression that uses only 8 and the plus sign that equals 1000." ... "The goal here is to get students to learn this in a way that lets them think flexibly about any base they work in, rather than just learning base 8."

More so that a direct approach to teaching about base units? Why not start with base 2, then 3, 4, and so forth? Then you could ask them why and when one base would work better than another? Why is octal used so much with computers? Why hexadecimal? How can you write hexadecimal? Why not have them learn to add and subtract octal or hexadecimal? The students might get a kick out of adding 7F and D5. How about converting from one base system to another. I taught my (to be) sixth grader about binary years ago and taught him how to convert to and from decimal. He was excited and loved to explain it to others. No discovery was necessary. There is no issue with his understanding.


With the 8's approach above, what, exactly, are the kids discovering? What, exactly, does "think flexibly" mean? Did you give them another problem of the same class and expect all of them to solve it? I am not a fan of indirect teaching and learning unless there is a carefully defined path. I don't see that path. However, it take more time and is neither necessary or sufficient. Discovery always has a nice aura, but what are the details?


"The students have been working on 'the train problem.' It uses five different train cars, with different colors and number of passengers it can hold. Students can use Cuisinaire rods to represent cars."

Why not use pencil and paper?


"Lots of amazing work by students in hypothesis and evidence, in use of mathematical principles."

What, exactly, are these "mathematical principles". I'm all in favor of students using fundamental mathematical principles to solve problems, but I don't see any of them mentioned here. What class of problems are you trying to get them to solve?


"Oh, after 30 minutes on the train problem, students spent an hour on fractions."

This is the necessary and sufficient part.


"For example, the 8s problem, noted below, was a struggle for the students, so as they tried to figure out how to think about it as a group."

Is this important? Group think?


"Today the kids will take a test, ..."

Can we see a copy of the test?


My impression is that you are way too concerned with process than with content and mastery. Perhaps you provide enough practice, but just find it more interesting to talk about some sort of vague discovery process. Most students are not rising math stars who might enjoy a class like this. (What do they have to compare it with?) Most students need a direct approach to content and mastery, grade-by-grade, that leads to algebra in eighth grade. It is not OK to let students get to fifth grade without mastering their times table. The problem is mastery of the basics, not some sort of vague idea of thinking flexibly. Master the basics and the thinking part will take care of itself.

July 20, 2007 at 7:24 AM
Catherine Johnson said...

Hi, Jenny!

What happened to your blog???

Do you know how it got hacked??

Should I be backing everything up on my hard drive?

(And thanks for posting!)

July 20, 2007 at 8:14 AM
Catherine Johnson said...

I'm at a very basic level here - I actually don't know what the phrase "rising 5th graders" means.

Does "rising" mean gifted, or does it refer to kids going into 5th grade (or kids going into 6th?)

July 20, 2007 at 8:17 AM
Jenny D. said...

The train problem was exactly the kind of problem faced by folks in operations research everyday. The Cuisinaire actually made the problem easier to work with for students.

One assumption made by commentors is that the teacher did NOT represent all of the work using properly mathematical notation. She did, and asked students to do it as well. Student created trains using the rods, but they were then asked to represent their trains using numerals, operators, and sometimes substituting rod colors, which would be actually be algebra, no?

Actually, the students figured out the 8s problem, and then were asked to represent various solutions in a table. Like how many 8s, how many 88s, and how many 888s could be used to create 1000. It was a puzzle, but one that required mathematical thinking and could be expressed with mathematical notation.

I don't think I can post the test. But if I find out how the students did I'll post that. Also, student today are presenting their work to observers and parents. One student just gave a fairly complex mathetical definition of fractions. He said define the "whole," break the whole into d parts. Then 1 over d is a fraction representing one piece of the whole. You can select n parts of d and represent it as n over d. Also, d cannot equal zero.

July 20, 2007 at 8:20 AM
Jenny D. said...

Rising 5th graders means kids who will be in 5th grade in the fall. Sorry for confusion.

PS. Blogger somehow brought me back from the dead. Can't really say more.

July 20, 2007 at 8:21 AM
Instructivist said...

"PS. Blogger somehow brought me back from the dead. Can't really say more."

This sounds intriguing. Shouldn't it be possible to say whether the blog was recovered from cache, give a hint of how the barbarian was able to break into the blog?

This is a frightening occurrence to all blog owners. Would a complex password have helped?

July 20, 2007 at 9:00 AM
Jenny D. said...

Got me. what happened was someone sent me an email saying my blog disappeared. I hadn't posted in a month, and when I looked some guy from Korea or something had taken my URL and blog name and started his own blog.

I went to my sign in and was told my blog was locked because Blogger thought it was a spam blog. I tried to unlock. meanwhile I emailed Blogger and left not-so-nice posts on the renegade jennyd site.

After about two weeks, Blogger restored by blog to its URL and unlocked it.

That's all I know.

July 20, 2007 at 12:24 PM
Me said...

I'd just like to second Barry and Steve's comments.

I can't even see what base 8 has to do with puzzle problem.

July 20, 2007 at 1:05 PM
SteveH said...

"I can't even see what base 8 has to do with puzzle problem."

I can't either. Can the students add the octal numbers 755 and 252? Can they convert the octal numbers to decimal and show that the sum is the same? If they can't, then what exactly are they learning to do? There seems to be a disconnect here. What are the "mathematical principles" that are being taught? How can these principles be used to solve other problems?

This relates to the issue of linkage between mastery of basic skills and "understanding". Many educators seem to think that understanding can exist outside of mastery of basic skills. They see mastery as only speed and not understanding. This leads to the belief that understanding can be achieved in a top-down fashion, with skills as almost a separate issue.

This is a very important point. How much can students understand about base systems without mastery of basic base system skills? Understanding without mastery of basic skills is rote understanding.

July 20, 2007 at 7:03 PM
Jenny D. said...

May I ask? How many of the commenters have taught fifth grade math?

I ask because it's clear that the commenters know math well. But unpacking that knowledge into building blocks that can be taught and then rebuilt is a different thing altogether.

Here's an example from literacy: I've had teachers say to me, "teaching phonics to 1st graders is really boring." Well, yes that's true. And the teachers may not fully appreciate the importance of teaching phonics because without it, you can't really read fluently and with understanding. So you have to start by teaching stuff that a college literature professor would find tedious.

This is probably true in math. As adults who are math fluent, we might not be able to know what it's like NOT to know math, and what it would take to get someone to learn math.

This is why expert mathematicians might not be the best teachers of fifth graders. It's why great athletes are not always great coaches. Because taking that expertise and turning it into building blocks for teaching is sometimes difficult for some who has achieved mastery and beyond.

July 21, 2007 at 3:59 AM
Barry Garelick said...

This is why expert mathematicians might not be the best teachers of fifth graders.

And great surgeons are probably not the best teachers for med school students, because they don't really know what, you know, pedagogy is all about.

I don't think expert mathematicians are claiming to take over 5th grade teachers jobs. They are saying that content (i.e., the building blocks to which you referred) are essential. And they know what content needs to be taught and the order in which it should be presented, and that it should build upon itself.

Teaching skills and content leads to procedural fluency. As Steve H says: "Master the basics and the thinking part will take care of itself."

The examples you gave from Ms. Ball's class are those in which she is claiming to unpack the building blocks of "flexible thinking". Problem solving and flexible thinking arise from domain knowledge and it's something that comes from experience. Challenging problems are one way, in a much larger package, to get at that, and as Wu indicated in what I quoted earlier, "puzzles and brainteasers" are one way to train for mental agility. But without some discussion of what it is they're actually doing with a particular brainteaser or puzzle, it becomes one more problem, and frequently (as Wu argues) becomes in the mind of the observer a solution that involves a "cute trick" of some sort.

People often point to the (possibly apochryphal) story about young Gauss who as punishment was told by his teacher to sum the numbers from 1 to 100. He comes up with a clever shortcut that yields the formula for the sum of n consecutive numbers. In an article I read, this story was used as an example of this "flexible thining" you're after and how most of us would simply add the numbers. Most kids would not come up with the solution that young Gauss did (and don't forget, he was a genius. Also, I don't know if the story is true). The premise of the author of the essay is that teachers are missing the boat because kids aren't learning to think flexibly.

The result is a widespread fad which has been embraced by NSF-EHR who has given millions of dollars to develop math programs based on the premise that giving kids challenging problems without the necessary and sufficient background to allow solving them will lead them to discover the tools they need to solve it.

The fact that Ms. Ball served as an advisor to the development of the second edition of TERC's Investigations makes me wary of techniques she holds dear, despite her skill in facilitating the discusion or her status as member of the National Math Advisory Panel.

To answer to your presumptuous and haughty question about whether any of us know-nothings teach fifth grade, as a matter of fact, I don't teach fifth graders. I teach my daughter, and I taught her and her friend math from Singapore's books in 5th and 6th grades when I could see that Everday Math was not providing her with the skills and content she needed to master. In my last comment here, I provided an example from the Singapore math book. It is a problem which I (and many others) consider to a be good, challenging, and proper problem in a 5th grade math class. I found that Singapore's program does what you recommend: It unpacks "that [math] knowledge into building blocks that can be taught and then rebuilt" and does so in a sequential and logical manner that results in mastery, conceptual understanding and (this may be hard for you to swallow) flexibility in thinking.

Like you, I am also in Ed School, although it isn't U of Michigan's, though I went to U of M to get my degree in math. I plan to teach math in middle school. Unlike you, I am not in a doctoral program, nor do I hold the theory that is taught in ed school dear, valuable, or useful. In fact, I find it destructive.

July 21, 2007 at 4:48 AM
Me said...

Thanks Barry!

As for credentials to "impress" Jenny, I have an M.A.T. and taught high school chemistry for several years (a long time ago) and even then could see that lack of math fluency had a huge impact on my students' ability to learn chemistry.

I (along with their father) taught my two sons math when they were little and paid close attention to the math they were taught in school. My younger son learned about binary numbers when he was a bit younger than Steve's son (described above) but I recall the same experience in that he enjoyed explaining it to other people because, you know, it made sense. He had a wonderful math teacher who (IIRC) explained binary fractions in terms of Cuisinaire rods that wouldn't fit through a door. This was in kindergarten or first grade.

BTW, I don't understand the current fascination with problems which only work with integers.

July 21, 2007 at 9:43 AM
Anonymous said...

While it is noble that your dean takes the idea of teaching and instruction (and teacher education) seriously, there is no data to show that just showing people ambitious instruction actually helps them transfer that teaching to their own context. Especially given that her teaching is curriculum-less. People need to actually try it out, see what the common mistakes are. Lots of research on p.d. and policy shows us that just watching isn't going to change instruction. Also, what is being done with the teachers of the students in the summer school class? Basically the dean gives them really good math instruction for two weeks and then....? They head back to their home schools and get more of the same low-level computation oriented instruction?

July 21, 2007 at 2:21 PM
Anonymous said...

Can anyone speak to their state's expectation for math achievement, or general expectations for public education? I'm curious if the "National Mathematics Advisory Panel" is aware of state constraints, such as cost-effectiveness implications of constitutional mandates for "thorough and efficient" education. My state's expectations are somewhat more elaborated than the panel's charge to "foster greater knowledge of and improved performance in mathematics among American students."

FWIW, over at JennyD's, I addressed Sherman Dorn's allegation that math-as-technology is inconsistent with teaching students to think flexibly, noted concerns with math teacher preparation, including that Singapore Gr. 6 Exam Is More Difficult Than the U.S. Elementary Teacher PRAXIS 2 Exam, and asked "Does math teacher preparation meet state constitutional standards?

The upshot of all this is whether teacher education meets the goals set for public education in their state--rather than just teacher college accreditation guidelines.

July 21, 2007 at 4:12 PM
SteveH said...

"As adults who are math fluent, we might not be able to know what it's like NOT to know math, and what it would take to get someone to learn math. This is why expert mathematicians might not be the best teachers of fifth graders"

Are you building an argument based on "might"? Is this a general argument or does it relate to something specific on this thread. You have to do better than that. I've heard this tidbit before.



"I ask because it's clear that the commenters know math well. But unpacking that knowledge into building blocks that can be taught and then rebuilt is a different thing altogether."

This is another vague, general argument. It's the old problem of arguing with generalities to make people go away so that educators can decide on the details. That's why we hear the idea of "balance" in math all of the time. Who can be against balance? I am. I want to know the details. Many of us have taught and have spent enormous time breaking down and packaging content and skills into lessons. We have lots of experience seeing what works and what doesn't. Don't treat us like we're stupid. Do you think these silly arguments are new to us? Do you think we are incapable of seeing and dealing with these issues. I'm tired of teachers lecturing me on things like "superficial knowledge".


This is my main point. Details matter. What are they. Educators can't just talk generalities and expect parents to oooh and aaah, but generalities are the only thing I got from the original post.

You don't tell us any details about the 8's problem or the train problem, but you say things like:

"...and I'm fascinated by how this group of observers talking with the teacher can offer different language and thought ideas to guide the students into seeing 88 as not tens and ones, but eights. This work is the basis for learning base 10, and base 8, and base whatever, which we may remember from our own math class."

and

"Lots of amazing work by students in hypothesis and evidence, in use of mathematical principles."


and then expect us to oooh and aaah. These descriptions are meaningless to me. What are the goals of the class? How do the lessons fit in with those goals? How does this course fit in with with the regular curriculum these kids get? When we ask specific questions, like what does "flexible thinking" mean, and what are the "mathematical principles", they are ignored. I'm tired of hearing about understanding and flexible thinking without anyone giving any kind of definition at any sort of level.

By the way, what heck are "thought ideas"?

July 21, 2007 at 7:34 PM
Anonymous said...

I'm tired of teachers lecturing me on things like "superficial knowledge"

But wait! There's more!
It seems the teachers' unions want to educate Supreme Court justices about "compelling governmental interest in educating all of our children to function effectively in a multiracial, democratic society and realize their full intellectual and academic potential."

So a reasonable question is, "how do education stakeholders in your state work together to fulfill the stated 'compelling governmental interest?'" For example, does the state board provide guidance to local boards so they fund only those professional development activities that serve the stated "compelling governmental interest?" Can professional development and curricular material providers demonstrate their products serve the "compelling governmental interest?"

July 21, 2007 at 7:57 PM
Jenny D. said...

I'll offer these thoughts:

In addition to thought experiments, these students did problems in a kind of "mad minute" drill. They answered simple arthimatic and multiplication facts, doing some number in a minute. A drill and kill exercise. Some of these skills were important to proving and answering various problems like 8s.

Some commentors have said that the 8s problem, for example, is not useful, that it would be more important for the students add the octal numbers 755 and 252 or convert the octal numbers to decimal and show that the sum is the same. You assume that none of these kind of skills were employed in the 8s problem. What if they were? Also, you can't know if a teacher trying to teach the addition of these octal numbers would be successful. What if students were taught that and didn't learn them? What if the real linchpin in math instruction is not as much what you teach but how you teach?

That's why this math class was important. It focused on how to teach. Remember, curriculum doesn't teach children; people do.

July 23, 2007 at 11:46 AM
Me said...

BTW, my first vice-principal told me that teachers (people) don't teach curricula, they teach children. This was probably my first clue that I was in the wrong field although I didn't realize it at the time. (I'm kind of a content-oriented person and ended up getting an advanced degree at the age of 36 and going into research.)

Anyway, I would really appreciate it if you could explain as simply as possible what the teacher intended for the children to learn from the 8's problem. I'm still confused. I know how to do the problem, I understand place value and octal numbers, etc., I just don't understand the teacher's intent in posing this problem.

July 23, 2007 at 2:17 PM
SteveH said...

"Some commentors have said that the 8s problem, for example, is not useful,"

One cannot tell because there were no details. Obviously, you thought your description was enough to generate ooohs and aaahs. It wasn't. When challenged, no more details came; just rhetorical questions.


"What if the real linchpin in math instruction is not as much what you teach but how you teach?"

What if it isn't? Are you guessing? This is one of the silliest statements I've ever heard.


"Remember, curriculum doesn't teach children; people do."

Nope. This is the silliest statement I've ever heard.

Generalities. Simplifications. Homilies. And no details.

Perhaps you don't understand that KTM exists SPECIFICALLY to counter this anti-content, anti-mastery, and anti-curriculum educational philosophy.

Education is not just one thing. Education has no process linchpin. There are many importatant parts and all of the details are important.

July 23, 2007 at 7:28 PM
Anonymous said...

Dammit, I missed this, and it's funnier than heck!

July 27, 2007 at 12:35 AM

Post a Comment

Subscribe to: Post Comments (Atom)