What’s Sophisticated about Elementary Mathematics? Plenty, writes Hung-Hsi Wu in American Educator.

. . . starting no later than fourth grade, math should be taught by math teachers (who teach only math). Teaching elementary math in a way that prepares students for algebra is more challenging than many people realize. Given the deep content knowledge that teaching math requires—not to mention the expertise that teaching reading demands—it’s time to reconsider the generalist elementary teacher’s role.

Many elementary teachers weren’t good in math and don’t like it.

In affluent, highly educated Palo Alto, 57 percent of parents provide extra help in math to their elementary school children, a survey found. Parents pay for Kumon classes, Score’s computer-assisted learning or other forms of tutoring. Despite protests from some parents, the district has adopted Everyday Math. There are reports more parents are hiring tutors to make sure their children learn the basics.

Yeah, my youngest daughter is realizing this. She is in a grad program to become a middle-school science teacher, and often has to explain the underlying math to other students. Can’t fault them, their colleges didn’t prepare them for the level of math they would need in life.

Can’t fault them, their colleges didn’t prepare them for the level of math they would need in life.Since when are colleges responsible for teaching students K-6 math?

I suspect that the quickest way to improve elementary math education in the US would be to shed our pride and adopt Kumon.

If elementary districts would stop adopting crap like “Everyday Math” or any of those other “rainbow and ponies” curriculum, that would be a start.

@Scrooge: They are not responsible for teaching K-6 math. But many students are able to graduate from college with little more than taking a cheesy math elective. Colleges could require all students to take some calculus or at least a rigorous statistics class. I would say most humanities/communication/arts students (many who become elementary teachers) end up taking classes that look like Investigation in Numbers for adults if they take any math at all.

I used to think like Scrooge until I started teaching a university math literacy course. It turns out there are many useful and sophisticated mathematical ideas that are accessible to 6th graders*, that only require the clever application of basic arithmetic. Adding columns of numbers is boring; adding columns of numbers to see if your team’s proposed solution to the traveling salesman problem is interesting.

It’s not that the colleges teach K-6 math, it’s that the colleges need to better prepare the K-6 math teachers, so they can teach kids math AND what math is FOR.

*and anyone else who is calculus-challenged

The difficulty argument is overstated and the (K-6) teacher math competence is understated. Look at the following site for sample NAEP test questions and results (4th an 8th grades).

Mathematics Report Card.

http://nationsreportcard.gov/math_2007/m0017.asp

If kids can’t answer these simple questions when they are in school for 6+ hours a day, then the problem is not difficulty, but competence.

The biggest problem I see is that K-8 educators know very little about math. It’s easy for them to get caught up in talk of understanding and critical thinking. Although they know that there should be “balance”, it’s the type of balance that never gets enforced. That’s why Everyday Math’s “trust the spiral” approach is loved so much. Schools don’t have to enforce mastery of the basics and they have pedagogical cover to claim that it really will happen. It doesn’t, or at least, it puts the onus on the kids and parents. Some kids do well, so it must be OK, right? Ask the parents of the good math students.

Another issue is full-inclusion. Math is cumulative and mastery of basic skills is paramount. These two things are incompatible. By the time schools get real and offer math tracks, it’s too late for many kids. But in many ways, tracking hides this problem rather than illuminates it.

I do see a change in teacher attitude about basic skills in 7th and 8th grades when our state requires that teachers have real content knowledge and skills themselves. If this were pushed back to lower grades, I would expect to see more pedagogical fireworks. It’s not clear who would win.

SteveH:

I think that you are spot on with regard to many issues here–specifically lack of teacher knowledge of how elementary mathematics lays a groundwork for later, more complex mathematics, and the trust in repetition at a later grade to pick up what has been missed. I would take issue with you as regards inclusion, however. My son was denied inclusion at nearly every point in his education. As a result, his mathematics (and all other content areas as well) was an incredible jumble. At many grade levels he was in broadly-graded and/or cross-categorical classrooms. Despite my constant efforts to get measurable and progressive goals written into his IEP, an IEP cannot take the place of a fully developed curriculum. If you think that a single teacher–even with as few as 12-16 students can manage to write curriculum for all subjects across multiple grade levels, and differentiate for all students, you are mistaken. And yet, this is far to frequently the state of the art for students who receive “special” education in a non-inclusive setting. Add to this the isolation from their regular ed teaching peers that most special education teachers experience and you can see that this impossible task is carried out without needed support from the folks who may in fact be content educated.

I can tell you how math began in every year from about fourth grade on. First there was a worksheet with some basic math facts (adding, subtraction, multiplication and division) hastily handed out on day one, to “get started” and “get to know” the kids. On the shocking discovery that the kids do poorly, they get more of these worksheets to learn to be able to produce more right answers–aided by matrices containing the right answer, or perhaps calculators.

My son “understands” addition and subtraction. That is, given a collection of objects, he knows when to remove and when to combine them. His most reliable means of getting the right answer is counting. As he is now legally an adult (and for the most recent six-eight years), I am not concerned with whether he ever memorizes these facts. I would like to see him gain increased facility with a calculator–because drawing and counting are painfully slow processes and not terribly accessible when standing in line at a store, nor practical for larger numbers.

But what concerns me most about this approach is that he has never really acquired much understanding of the meaning of multiplication or division, or how they are useful, ditto fractions, decimals, etc. And, despite his disability, his IQ is in the normal range. As a participant observer in this process it has been endlessly frustrating that there is seldom anyone with whom I can even have a conversation about the whats and whys of mathematics–beyond “he’ll need it later in life,” and a firm conviction that either faulty attempts to drill and memorize facts, or to repeat over and over again the same worksheets, would provide a new and different answer some year. I researched and brought in lesson plans, manipulatives, even a whole very interesting set of lessons based on using spreadsheets.

Suffice to say, I think that inclusion is a red herring. If we were equipped to ensure concept mastery for students–and to intervene promptly when they don’t get it (as opposed to believing they’ll have another chance next year and the year after), I think that not only would we stand a far better chance with the students that we would rather exclude, but for all students.

>>Colleges could require all students to take some calculus or at >>least a rigorous statistics class.

They could, and there would be some benefit to this. But – tying into the earlier post – calculus is the not the kind of math people need for their everyday life.

Basic (but rigorous!) statistics – say up to and including an understanding of standard deviation, but not including regression analysis – would be useful for most people, however.

For some teachers, a basic knowledge of fractions, rates, and how to calculate percentages would be an improvement. I sometimes think that many adults never really understood fractions, and skated by in the rest of math.

Sorry, but a lot of these educator are full of crap. K-6 “math” should consist of arithmetic, fractions, and decimals. You don’t need to have majored in math to teach this stuff. My wife, back in her elementary teaching days, was an excellent math teacher because she had so much trouble with it in high school, so she knew from first-hand experience where her kids were having problems. (But I tend to agree that high school teachers should have some familiarity with college math before teaching high school math, but that’s not to say that they need to have majored in it. My degree was in engineering, and I’ve forgotten more math than is ever taught in high school, but because I didn’t major in math, for some reason I am considered unqualified to teach high school math! Insane!)

As for this “new math” notion of learning math nomenclature in elementary school, first, it’s not necessary, and second, it detracts from learning the required skills and becomes a tested end in itself. Learning sets in high school is good enough for students whether they are going on to college or not. There is no reason on earth why K-6’s should be exposed to set theory to the exclusion of fractions.

Peter W,

People “need” calculus just as much as they “need” a foreign language or “need” English literature.

Sure, it’s hard for some people, but not much harder than foreign languages or writing about literature was for me.

There used to be a time in college history when you couldn’t receive a degree without taking and passing calculus. However, since the 1980’s, the quality of education has decreased, and the costs have increased about 400%.

If you look at actual classroom instruction time, students in the 1950’s, 60’s, 70’s, and though most of the 80’s met for a much longer period of time than they do today.

Want to see for yourself, check a major college campus class schedule, you’ll find almost no classes scheduled on a friday, most classes are 75 minutes twice a week (2.5 hours of instruction).

The amount of homework handed out in introductory classes is somewhat lacking as well, as your typical student taking english 101 in the 60’s and 70’s would probably write 15-20 papers in a semester, students today would probably write less than 10.

A college professor once remarked on how standards declined, as a textbook he once used for an undergraduate course is now being used in graduate level studies.

Makes one wonder how valuable college today really is.

I think the problem is that math teaching is chock full of bad ideas. If you want a sample, go over and look at the Donor’s Choose website. Recently, I thought it was time for me to donate again and I went to Donor’s Choose and thought, “hey, I like math, I’ll donate to a math project.”

What I found was disgusting. The most common need teachers posted was for graphing calculators (I don’t think they should be allowed in classrooms before college). There were requests for all sorts of games and “tactile” this and “group” that. Hell, there was even a request for “math journals”. I had to google around a bit to find that this is evidently very vogue in math teaching these days. I can’t see how writing about studying math can possibly make you substantially better at math. Maybe it helps with English, but as it comes at the expense of your math time, it’s goofy.

Nobody was asking for basic tools, like a compass that can be used to draw on the chalkboard or number lines or protractors for the kids. I had to give it up and fund a history project, instead.

It seems like it would be pretty easy -just by looking at test scores- to show that all of this junk doesn’t lead to students who understand math better. Why can’t we get rid of all this junk?

By the way, when it comes to how much math you “need” in your everyday life I think people kind of miss the point. You’ll never know how math could have helped you in a situation if you never learned the math in the first place. So if you’ve never learned much math, you will always conclude that you never needed much math.

I’m a software guy for a company that does a fair amount of engineering. Without calculus, I just wouldn’t be able to understand a fair amount of what the engineers tell me. Obviously, I couldn’t have predicted this way back at the time I was in college.

Robert Heinlein used to claim that anyone who didn’t know math “wasn’t fully human”. That’s not an attitude that would be very popular today, I suppose.

“You don’t need to have majored in math to teach this stuff.”

For K-6, this is true, and I don’t like the spin of the article that says that you need something special or math specialists in K-6 to prepare kids for algebra. It’s not that complex.

K-6 teachers need more math, but what, exactly, does that mean? It doesn’t mean that there are some special or complex math teaching skills that are missing. What is missing is an appreciation that mastery is not a rote process. There is linkage between mastery and understanding, and there are many levels of understanding tied with mastery. This is so obvious to those who have any math background, and painfully missing from those who don’t.

So what happens? In K-6, kids are exposed to math as some sort of mystical thinkng process that can’t be broken down into specific skills. (Anti-Math) Math curricula go out of their way to find problems that don’t have one solution as if to prove this point. Forget the fact that guess and check fails completely when the numbers aren’t nice. I like to say that math allows you to think less. I also like to tell kids to let the math give you the understanding. Mastery is more than just speed. It’s a foundation for understanding. Understanding is encoded in the skills.

But it’s really more basic than this. Look at the NAEP link I gave before and the sample problems and results. Schools don’t seem to take any responsibility to ensure any level of mastery. If you think that your job is just to lead the horse to water, then you have no way to judge whether what you do is good or bad. Many just assume that if you go over the material enough times, then mastery will be achieved, but you can’t trust the spiral. If you do, then it will be “game over” by seventh grade.

“I would take issue with you as regards inclusion, however. My son was denied inclusion at nearly every point in his education. As a result, his mathematics (and all other content areas as well) was an incredible jumble.”

Then you are talking about some sort of inclusion I’ve never seen. At my son’s school, full inclusion means having a range of abilities so wide in one classroom that the the teacher can’t possibly get it to work. If kids don’t master basic skills, they have no way of determining whether it’s because the kids are “not ready” for the material yet, or whether they are doing a bad job of teaching. Most likely, the teacher is trying desperately just to keep all of the kids busy. Nobody is on the same page and there is no way to ensure anything. In our schools, full-inclusion means placing the onus on the kids and parents.

One charter school in our area is taking a different approach. Instead of full-inclusion classes, they have a full-inclusion environment. The core academic classes are filled homogeneoulsy, but everything else is mixed ability. Teachers aren’t spread thin trying to be ten different things to 10 different groups of kids in one class. In our full-inclusion school, they use differentiated instruction, which is some sort of magical technique where each child can get what they need even if the kids are in child-centered, mixed ability groups.

Pedagogy first, reality second.

“Then you are talking about some sort of inclusion I’ve never seen. At my son’s school, full inclusion means having a range of abilities so wide in one classroom that the the teacher can’t possibly get it to work. If kids don’t master basic skills, they have no way of determining whether it’s because the kids are “not ready” for the material yet, or whether they are doing a bad job of teaching. Most likely, the teacher is trying desperately just to keep all of the kids busy. Nobody is on the same page and there is no way to ensure anything. In our schools, full-inclusion means placing the onus on the kids and parents.”

Steve–what I described was not inclusion–it is the EXCLUSION that my son experienced. And it is nearly identical to your description above. The “resource room” has a far wider range of abilities, ages and achievement than is tolerated in the “regular” classroom. In my district, “inclusion” is reserved for kids who can “survive” in the regular classroom with no (or minimal) additional support. In the “resource room,” the primary means of “support” is “smaller classroom” and “specially trained teacher.” The specially trained teacher means one with general knowledge of disabilities, accommodations, etc–but far less content training than a regular classroom teacher. One year my son received “one-on-one tutoring” in math. It turns out that “one-on-one” tutoring actually was three students and one adult in a closet for a short time each day. The three students were actually being tutored on different subjects. The tutor was trained in reading.

But – tying into the earlier post – calculus is the not the kind of math people need for their everyday life.The advantage though of trainee elementary school teachers learning calculus, or some other advanced maths course, is that they would get an idea of where some of their students will eventually be going with maths. For example, NZ school teachers decided to stop teaching long division by hand as “kids could always do it with calculators”, not realising that universities teach polynomial division assuming that the incoming students have learnt the ordinary long division algorithms. More generally, doing calculus requires a good knowledge of algebra which in turn requires a good knowledge of fractions. That can emphasis the importance of teaching fractions thoroughly in the classroom.

On top of this, one key advantage of studying advanced maths is that it really expands your understanding of the pre-requisite maths and your experience at doing it. If you learning calculus, while the foremost area of your brain is engaged with learning calculus, you are still practising algebra and driving that deeper and deeper into your brain, in a more interesting and engaging way than just doing pure algebra exercises over and over again.

Finally, from my experience tutoring, I am inclined to think that teaching a skill well (be that solving word problems using algebra, or downhill skiing) requires greater knowledge than merely using the skill. Teachers, at least really effective teachers, really need to know their stuff more than the average person does in their daily life. I hope an experienced teacher will correct me if I am wrong about this.

SteveH and Margo/Mom, you’re both illustrating the need to base instruction in math (and other subjects) on correct identification of a child’s zone of proximal development (i.e. what they’re ready to learn next, which is based on assessment of what they’ve already mastered) and then placement in whatever classroom setting will deliver instruciton in that ZPD. I would add, it’s best if the classrooms are also characterized as to whether they will move quickly, at average pace, or slowly through the upcoming material but that’s quite a lot to hope for. I made it through College Algebra/Trig in high school, but only because the pace was quite moderate. I needed a lot or practice.

“…you’re both illustrating the need to base instruction in math (and other subjects) on correct identification of a child’s zone of proximal development (i.e. what they’re ready to learn next,..”

No. We’re both illustrating how screwed up many schools are. In our school, they claim that differentiated instruction allows the ZPD to be achieved. Only in their dreams. It’s also not just about ZPD. It’s about what constitutes a proper education. It’s about low versus high expectations. It’s about ensuring mastery, not depending on ZPD and letting nature take its course.

By the way, “College Algebra/Trig” is an oxymoron. Algebra and trigonometry are high school subjects.

SteveH, read what I said. I indicated that children should be placed in classrooms (i.e.grouped) according to ZPD, not that they should recieve “differentiation” that allegedly will allow them to progress in their ZPD. And, not to worry, College Albegra/Trig really is the name of a high school course. As it was in the ’60’s when I took it. Although it’s also taught in plenty of colleges today. In other words, Steve, I’m agreeing with much of what you say about your frustrations about how math is taught these days.

Jane and SteveH: There are two ways to differentiate. One way is to differentiate content. This is the basis on which students have been tracked, grouped and otherwise separated. Students are actually learning different things, based on different sets of expectations. And–from whatever point this differentiation begins, their lot tends to be set.

Differentiating instruction is something else again. The content and expectations are the same, but the methodology of teaching is different. It is easier to think about in some cases than others. A student who is blind would read in braille, or use a reader or tape recorded text. but the content that the student learns would track along with the rest of the class. We have tended to think of students with disabilities as being “slower” or “behind” their peers. This is frequently not the case, or need not be the case–the majority of students with disabilities have an average range of intelligence–but experience a range of barriers, from physical impairments to attentional difficulties to mismatched emotional and intellectual development.

I am familiar with Vygotsky and the ZPD–however I don’t believe (and suspect that Vygotsky wouldn’t either) that this is something that is set, hardwired and procedes at an immutable pace dependent on the individual child. Yes it is important that instruction take into account that zone. What was revolutionary about Vygotsky’s work was that he challenged the presumption that what had already been learned and could be demonstrated independently was the only important indicator. The ZPD actually defined the current work of the child–that is the area in which they are actively learning. One might assume that without attention to instruction in that zone, the child might never move forward, or that the zone might actually shrink. This might provide an argument for flexible grouping with continuous assessment and immediate intervention. It is hard for me to see it supporting classroom-based sorting, or the exclusion of groups of students based on disability status.

No argument there, Margo. But SteveH was concerned about classrooms where (even assuming the children had been assessed), there was no flexible grouping, just putting all children of the same age in the same classroom, regardless of what they were currently able to learn or ready to learn next. And, your example of blind students is not really to the point. Using “grade level” as a rough indicator of what the student is learning now or ready to learn next, many blind students can move along throught the curriculum (with the use of technology) at the same rate as the average student, or better. Many children with other types of disabilities or just slower learners or those who’ve missed a lot of school or some other complicating factor, can also move through the curriculum, but not if they are forced to skip over what they haven’t learned yet. Teachers who are differentiating (in the sense that use the word) for many different levels of students may provide some instruction to each child, but they do not provide a coherent curriculum to each child, on her or his level; they basically have to focus on the middle of the range.

Your son (and my daughter) didn’t recieve good math instruction but to my mind it’s not because they weren’t in regular classrooms (at least for my daughter, who is also of normal intelligence). It was because the classrooms they were in were just as multi-level as the regular classrooms were. Multi-level doesn’t work in regular ed, and it doesn’t work very well in special ed either.

I think classroom hours need to be increased. Other countries have students being taught in the classroom for much longer times then in the states. You can see the difference in the test scores world wide. I think President Obama is considering additional hours for students.