# Algebra starts early

Teaching algebra starts in elementary school, Los Angeles has decided. From the Daily News:

WEST HILLS — Playing a game with crayons and colored blocks, 6-year-old Skylor Bates looks like he’s enjoying recess instead of learning algebra.

But the game’s patterns introduce Skylor and his first-grade classmates at Pomelo Drive Elementary to the concept of abstract thought — one of the keys to understanding algebraic equations.

Ninety percent of Los Angeles students score below proficient on the state’s algebra test. Students must pass basic algebra to earn a high school diploma. It’s not just a question of learning to go from the concrete to the abstract.

Educators said students also struggle because they have a weak foundation in other fundamentals, such as fractions, decimals and percentages.

Here’s my favorite quote:

After a recent class on patterns, 7-year-old Mason Bissada proclaimed: “When I heard that that was algebra, I was like, oh, I did that in kindergarten . . . It makes a lot of sense.”

I’ve been doing a lot of algebra tutoring with ninth graders this year. I wish they knew addition, multiplication, fractions, decimals, percentages and negative numbers. And I wish they had the faith that this is all supposed to make sense.

1. Steve LaBonne says:

“Educators said students also struggle because they have a weak foundation in other fundamentals, such as fractions, decimals and percentages.” Well duh, it took them all these years of screwing up kids with new-new-math foolishness before they were able to attain this great insight? Good Lord.

2. BigFire says:

For some reason, I get the feeling that if I were to have children, I must school them Math myself. The most useful mathematic lesson I ever learn was the memorization of multiplication table in the 4th grade in Taiwan. After that, all math until Calculus was easy.

3. cwlives says:

I have to echo Steve’s comments. Last year (7th grade) they started giving timed multiplication and division math tests using the integers 0-9 for multiplication (and the corresponding numbers for division). It was 60 questions and 4 minutes maximum time. Easily > 60% of the students could not come close to completing the test. Thankfully I saw this problem a long time ago and got my daughter tutoring (patience problem for ol’ Dad here!).

The principal is an ex-Marine, this year they have twice as many hours of math instruction, only one middle school math teacher returned AND one of the added teachers is an engineer! A big step in the right direction.

Chris

4. ricki says:

Are kindergarten students adept enough at abstract thinking for even the BEGINNINGS of algebra to be worthwhile?

I’d much rather see kids get a good grasp of basic arithmetic, and then when they’ve mastered that, do algebra.

I know in some respects I wasn’t “ready” for algebra even though I took it in 7th grade. I had to do a lot of re-teaching myself the basics of it when I was in high school.

5. Steve says:

“But the game’s patterns introduce Skylor and his first-grade classmates at Pomelo Drive Elementary to the concept of abstract thought — one of the keys to understanding algebraic equations.”

Algebra is NOT abstract! It is very concrete and very clear in its framework and rules. You do not use patterns to understand or solve algebraic equations. Telling kids that finding patterns will help them learn algebra later on is ignorant and delusional. It takes time away from the real need to learn and master the fundamentals of arithmetic – add, subtract, multiply, divide, ratios, percentages, fractions, simple word problems using distance, rate, and time, expressions, equations. (without a calculator, graphing or otherwise!) All of these subjects are very concrete and have little to do with patterns. This fascination with abstract thinking and finding patterns comes from people who do not know math, but pretend to know how to teach it.

My son has had this “find the pattern” stuff over and over since Kindergarten. Tangrams anyone? Do you notice that all of this pattern searching stuff requires no rules or previously learned skills? This goes along with the progressive thinking that math is this strange, abstract world which requires some magic zen-like ability to come up with solutions without having any background knowledge and skills. Kind of like the modern equivalent of creative writing – education without basic knowledge and skills.

Students do poorly in algebra because they aren’t properly prepared in the basics. I taught algebra and trig. in college for many years. Perhaps my students thought that the material was abstract, but the problem was that they just hadn’t learned and mastered the basics. If you have big gaps in the basics, then everything will seem confusing and abstract. My students would try to *pattern* one solution after another, without any idea of what they were doing. I suppose that if you are lacking in the basics, then searching for patterns is all you can do.

I want to know who decides on the math curriculum for the lower grades. Is it the algebra teachers (and math department of the high school), who presumably know something about math, or is it the administrators and teachers in the lower schools? At our public schools, it is the latter. Do high school math teachers complain that the kids don’t know the basics, or do they complain that they just don’t know how to find patterns? I have mentioned in the past how our lower schools know that there is a curriculum disconnect between K-8 and the high school, but they don’t appear willing to go to the high school and ask them what they need to do to fix it. The high school teachers also appear unable or unwilling to go to the K-8 schools and tell them that there is a problem. Is this a problem with academic turf? Isn’t everyone on the same team?

People who decide on math curricula need to know something about the subject. Apparently, this isn’t the case. I can just envision a group of lower school teachers and administrators in their curriculum task force meeting talking about how they all did not like or do well in math. It couldn’t be because they weren’t taught the basics properly, it has to be because math is so abstract. Maybe that is why many schools teach fuzzy math.

Math is concrete, these educators are the fuzzy ones.

6. mike from oregon says:

I read the actual article and didn’t quite see what they were/are doing differently. In fact, from experience and from the comments that I read here it really sounds like it’s time for some ‘old school’. Chris mentioned that his child’s school tested on multiplication and division tables – duh – my mom (and in turn I drilled my kids) use to have ‘races’ with kids in the neighborhood with flash cards. To be able to do the flashcards we had to know and memorize our addition, subtraction and especially multiplication and division tables. We would often drill sitting in the car waiting for one of my siblings to come or go from someplace. Kids in my neighborhood HATED to join us in the car (my mom would sometimes bribe them into the car with money and would put up a money ‘prize’ for the person who answered the most cards fastest and correctly. Needless to say, we smoked them. As a kid, I didn’t much like the flashcards, as I got older, it turned out to be one of the best things she did for us in our young education.

I say to hell with all this ‘new’ ways of learning (including english) get back to old school. It has years of results behind it. Bottom line solution to one of the main problems – REQUIRE the kids to do the work and pass at a certain level (like 70% minimum) and only THEN allow them to pass to the next ‘grade’.

7. We’re coming up against the same thing.

I’ve got two kids that are trying to solve word problems that call for multiplication and division – without having gotten a firm grasp of how to multiply, never mind memorizing any multiplication facts. One day, I see addition word problems for homework, then suddenly there’s multiplication word problems without any simple multiplication drills in between.

I thought that was quite odd. I’ve got them drilling on multiplication facts, because they’ll work too slowly and inaccurately otherwise, and they certainly don’t have enough fingers to help them out! I wonder if their school’s curriculum guide is missing a few pages…

At any rate, elementary math does itself include a fair bit of algebra. No solving for X, but to multiply multi-digit numbers, the standard algorithm they teach in elementary school relies on two algebraic facts:

a * (10 * b) = 10 * (a * b)
(a+b)* c = a*c + b*c

No one mentions that at the time, and you see it years later in an Algebra class without the connection being drawn to your attention, or at least that was the case when I went to school.

8. I’ve been ruminating lately on the idea that the reason there seems to be two distinct mind-sets on everything remotely political is that what passes for “thought” in roughly half the population consists of emotional reaction, associative recall and pattern recognition, while the other half relies more on logical deduction and cause-and-effect. Right-brained versus left-brained, in other words.

Not that either approach is necessarily bad if applied to appropriate topics, but I can’t see much practical use for “algebra by and for poets”.

9. Steve – I have been a high school math teacher for 14 years now. I have taught every thing from Basic Algebra through AP Calculus. I don’t know where you got your ideas, but Algebra is very abstract! Right now I am teaching a class of students what our school calls extended algebra. The state of Texas has decided in all of its infinite wisdom that no math below Algebra should be taught at the high school level. So the kids who have failed math (some every year since the 4th grade) have to jump right into Algebra I, and they are not ready. Yes, they lack many of the basic skills, but we put a calculator in their hand to get around that. Even with the calculator help, they still don’t get it! And the reason they don’t is that their minds aren’t ready for the abstract thought processes necessary for Algebra! Calling x a number and mathematically combining it with other x’s takes the ability to see in the abstract. There are no two ways around it. Some kids understand this in 6th grade; some don’t get it until late in high school, and some never get it at all. I have a few kids in class who are taking it for the second or third time, and they are finally getting it, not because they have memorized the patterns, but because their brains are finally able to think on an abstract level. If you know anything about Bloom’s Taxonomy, you can see that Algebra reaches up into the higher level thinking skills more than math ever has up to this point.

Do our kids need more drill in the basics? Yes! Will patterns help them in Algebra? Yes, somewhat. Should we sacrifice time spent on introducing patterns in kindergarten when we need to spend more time on other things? I don’t know.

And to the question on who writes curriculum – it is written by a combination of teachers and administrators. Math is changing – technology has made it change. We can’t completely go back to “the way it used to be when we were in school,” but we shouldn’t throw out all of the old ways of doing things. There needs to be a balance.

10. lindenen says:

How has technology made math change? This is one I’m dying to have explained to me.

11. 1)Re the “is algebra abstract?” question…the *operations* of algebra are concrete; the *concepts behind them* are abstract. The idea of manipulating a variable *without knowing what it stands for* is probably a difficult one for many people to grasp.
2)Neurologists and experimental psychologists probably have something to say about the age range at which abstract thought capabilities develop. I wonder if the authors of this curriculum consulted any such research.
3)Jill…could you explain how you think technology has made math change?

12. Bluemount says:

I always liked math because there was a ‘right’ answer, unlike more subjective subjects that were often based on opinion. If the math concepts led to abstactions it was because reality itself is filled with abstraction. Although we we were allowed to use a calculator when I took more advanced math courses, we were taught on a sliderule. The teacher demonstrated repeatedly he could find the answer with a slide rule faster than anyone using a scientific calculator. I don’t remember how to use a sliderule, but it wasn’t hard. I don’t think it’s good to believe we are so helpless without technology. I thought math was about understanding why things work.

I think very small children would benefit from understanding patterns that relate to real symbols. But only if it a small example in routine exercise. I thought there were some very nice examples of this in some computer games like ‘Treasure Math Storm’. It didn’t replace exercise, but it was a fun and skill building diversion.

13. Steve says:

Jill wrote:

>”Yes, they lack many of the basic skills, but we put a calculator in their hand to get around that. Even with the calculator help, they still don’t get it! And the reason they don’t is that their minds aren’t ready for the abstract thought processes necessary for Algebra! Calling x a number and mathematically combining it with other x’s takes the ability to see in the abstract.”

I guess it depends on what you call abstract. Calculators do not make up for a lack of understanding of the basics. You are saying that because you give them calculators and they still have trouble, then it must be because of the abstraction. I don’t agree with that. Introducing variables into equations is not a very big jump of abstraction. The rules of manipulating equations are no more abstract than the rules of Monopoly. Perhaps if you don’t know how to manipulate fractions, then algebra seems abstract. If you don’t practice, then algebra will always seem abstract.

Distance = rate * time (D=R*T)

I need to get to Boston in two hours and Boston is 120 miles away. How fast do I have to go to get there?

120 = R * 2

How abstract is that? One equation in one unknown. Solve for R.

The tax rate in our state is 7 percent.

Tax = Sales Tax Rate*Price (T = .07*P)

How abstract is this? One equation in two unknowns. What is the tax on something that costs \$150? Plot this equation on graph paper to show the students how useful this kind of equation can be. Explanations of more complicated equations are built on explanations of simple ones. Difficult perhaps, but not abstract.

Math is cumulative. When it is presented properly and built on a solid foundation of the basics, it is not abstract at all. Some students might not like it or might have difficulty with algebra, but how can you say that it is difficult because it is abstract. In all of my college teaching of math, abstraction was never the problem, lack of understanding and mastery of the basics was the problem.

>”If you know anything about Bloom’s Taxonomy, you can see that Algebra reaches up into the higher level thinking skills more than math ever has up to this point.”

Are you saying that there is a big jump that takes place here? Says who? Is this jump due to abstraction, whatever the heck that really is? By the way, what exactly are “higher level thinking skills”? Those skills that depend on numerous lower-level skills (the basics)? Is this supposed to be the same as abstraction? I really hate vague terminology.

>”I have a few kids in class who are taking it for the second or third time, and they are finally getting it, not because they have memorized the patterns, but because their brains are finally able to think on an abstract level.”

How do you know this? What do you mean by the ability to think on an “abstract level”? Could it be that that they just finally got enough of the basics that they missed in the lower grades? Having a student finally “get it” doesn’t mean that the material was abstract for them. If you label it as some sort of individual vague abstraction problem, then you end up with vague solutions, like teaching kids patterns in the lower grades – with no guarantee that it will solve anything.

By the way, why don’t you go back to the lower schools and tell them not to send up students who are not properly prepared in the basics?

>”And to the question on who writes curriculum – it is written by a combination of teachers and administrators. Math is changing – technology has made it change. We can’t completely go back to “the way it used to be when we were in school,” but we shouldn’t throw out all of the old ways of doing things. There needs to be a balance.”

.. Combination of teachers and administrors… I said that, but do the people who select the K-8 math curriculua know anything about math? Do you or any of the other high school math teachers get any say in the K-8 math curriculua? If not, why?

Technology is changing math? No, math is math. People can use calculators and computers to solve more sophisticated problems than before, but a system of equations is still a system of equations. How much should students use calculators and computers in the learning process depends on a number of factors, but I see great distraction and little help in using these tools in teaching the concepts and skills of algebra.

As for “balance”. I don’t like this word. It is used by those who do not want a return to an expectation of mastery of basic skills and knowledge.

14. David Gillies says:

Lindenen et al. – how has technology made maths change? Quite a bit. The obvious way is the introduction of the calculator. This enables you to much more rapidly and accurately do things that were really quite hard before. No need to grab a slide rule or log table to calculate the fifth root of 20, for example. What it does not replace is the need to know when the answer you get is bogus, and that is a very hard skill to teach (it was drummed into us repeatedly while I was doing my Physics degree, and it’s saved my neck on many an occasion).

Computers go one stage further. I have an amazing program called Mathematica on my computer. With it, I can do experimental maths, and find solutions to problems that are very nearly intractable for all but expert mathematicians. There’s nothing like playing around with the parameters of an equation to get a feel for it, and that is what differentiates mathematical insight from mere number crunching. To actually see what happens to, say, an inequality plot in the complex plane as you shift one of the terms and make a movie out of it is incredibly powerful. Funnily enough, Mathematica isn’t really an awful lot of help as a cheating aid. It will give you the final answer, but if you want to show your working you’ll have to go through the stages like everyone else. Building these stages and plugging them in is no different from working through a paper proof, and just as enlightening.

15. hardlyb says:

Of course technology doesn’t change the correctness of a given mathematical statement, but it obviously does make a big difference in how people do calculations, and the sort of calculations that people can do. And it also changes, in some cases, the kinds of problems that mathematicians work on. An example of this can be found here:

http://math.stanford.edu/comptop/

It can also change the way math is taught. My oldest kid takes math classes on the computer from EPGY:

http://www-epgy.stanford.edu/

But introducing “technology” into math instruction has almost always been a disaster in my experience, because the mathematics that most pre-college students need to learn is hundreds or even thousands of years old, and was done without the use of anything other than a pencil and paper. The EPGY people actually know math, so their use of the computer is aimed at teaching kids who can’t be in a classroom, and so their software tries to make up for some of the missing interaction with a teacher.

I have never seen calculators used in a math class in a way that didn’t detract from the class.

16. Steve LaBonne says:

It should not automatically be assumed that technology should change the way math is taught. It’s very dangerous to use the crutch before you really have a firm grasp of what it’s doing. Mastery of paper and pencil calculation should come first. As a former college professor I’ve seent the results of violating that maxim- students from “good” schools majoring in science at a good college, who are mystified by the simplest algebraic manipulations, who can’t estimate to save their lives or recognize whether an answer is even in the right ballpark or not, and who have no grasp of the concept of significant figures. Teachers, please THINK, instead of just repeating mindless slogans (many implanted in you for their own benefit by the manufacturers of calculators and computers!) about how technology has supposedly “changed math”.

17. Steve says:

I started college during the transition from slide rules to calculators. It really improved education. It didn’t make it easier. The five page hand calculation homeworks were replaced with 30-40 page calculator homeworks using sophisticated techniques that could not be used if they were done by hand. However, we already knew algebra and the basics. The exams changed from problems with numeric numbers to problems with variables that we had to manipulate algebraically.

The problem is that many educators think they can use graphing calculators and computers to reduce or eliminate the need to learn basic knowledge and skills. Programs like Maple and Mathematica can be useful in education as an add-on, but not as a replacement for the mastery of algebraic manipulation. Achieving this mastery requires a lot of hard work and practice. These programs and calculators provide no substitute for this process. In fact, they get in the way.

18. Jill says:

Technology has changed mathematics in that we are now able as teachers to explore in the classroom problems that without a calculator would have taken hours. We can stop spending time on problems with contrived data that have no relevance to the real world and actually take real-world data, determine its trends (and they are not always linear) and use numbers from real-world situations. So many math books (even from my day, and I graduated in 1983) have problems that have very little relevance to the average high school student. But with a calculator at hand, you can explore problems with ugly numbers and not get bogged down in the number-crunching. I started teaching without graphing calculators – now I teach with them constantly, and I love the changes that they bring. Now we also do not have to spend an entire week teaching kids how to read a log table or a trig table – we can spend that week doing real-world problems using logarithms or trigonometric functions.

Steve – back to your comment about abstractness, since your experience comes from teaching at the college level, you of course would not encounter much lack of understanding of abstract thinking, because your students are older. I have a student, for example, who can manipulate numbers just fine, but when I ask him if I know that total number of games a team played and how many the team lost, he has no clue how to find out how many games the team won. That is an abstraction problem. Also, my kids who are taking the course for the second and third time finally get it, but some of them still can’t add and subtract positives and negatives without a calculator. This is an abtraction issue as well. Symbolic manipulation and abstract thinking are both foundations for a successful understanding of algebra. The first can be overcome (to a point) with a calculator – the other must be acquired by maturity.

19. Jill says:

PS – I teach at the high school level – calculators should not be used at all before this point, except under the rarest of circumstances.

20. Steve LaBonne says:

Jill- thanks for that last comment, which is very reassuring. Still, the use of graphing calculators even at college level needs to be carefully watched- it can all to easily become a substitute for real understanding, as I’ve seen firsthand. “Number crunching” may not be a sufficient condition for developing “number sense”, a feel for how equations behave, and other aspects of genuine mathematical understanding- but I have seen absolutely no convincing evidence or argument that it is not a necessary condition. I don’t think any art school worth its salt would immediately start teaching, say, the making of video installations to beginning students without some training in the traditional hands-on skills like drawing. At least I hope not.

21. Mike in Texas says:

Jill wrote

If you know anything about Bloom’s Taxonomy, you can see that Algebra reaches up into the higher level thinking skills more than math ever has up to this point.

Its interesting you should bring up Bloom’s taxonomy as it is the misapplication of Bloom’s taxonomy, the need to push for higher order thinking skills, that has created a generation of kids who don’t know the basics. I would love to see what a kindergarten teacher has to say about the higher order thinking skills of 5 year olds. When you are wasting your time trying to push things that are not developmentally appropriate for the child’s age things like the basics tend to get pushed out. Another consequence of high stakes testing and ultimately what you get when you let politicians make the education rules.

22. Steve LaBonne says:

Interesting non-sequitur in that last sentence, especially since many other teachers and educrats, who have swallowed the “higher-order thinking” Kool-Aid, like to blame “high-stakes testing” for making them spend too much time on- wait for it- the basics!

23. Steve says:

Jill wrote:

>”I have a student, for example, who can manipulate numbers just fine, but when I ask him if I know that total number of games a team played and how many the team lost, he has no clue how to find out how many games the team won. That is an abstraction problem.”

Why do you call this an abstraction issue? Why isn’t this simply a matter of teaching and practice? Kids (even college level) run into all sorts of misunderstandings in math. How do you decide if it is an abstraction issue or something else? As a teacher, you shouldn’t care. You have to try whatever you can to figure out the problem and resolve it. You are labeling this an abstraction problem that can only be solved by maturity. This sounds like an excuse for not trying. If you come to this conclusion after spending a long time with the student, that is one thing. However, I don’t like vague terms like abstraction and maturity being used to explain away all sorts of failures in basic math education. I have seen nothing that suggests that abstraction is even a very small reason why kids do poorly in math. And, I don’t like educators, who don’t have a clue about math, using vague or popular concepts, experimenting on our kids.

>”Also, my kids who are taking the course for the second and third time finally get it, but some of them still can’t add and subtract positives and negatives without a calculator. This is an abtraction issue as well.”

Well, then why bother to teach them. If it is a maturity problem, then the second and third time you teach them doesn’t matter. Just give them an Abstraction Test and wait until they are mature enough to pass it. Could it be, however, that the each time they take the class they learn a little bit more of the basic knowledge and skills they missed – until they finally “get it”. Then again, maturity has to do with learning and practice, not time or age, doesn’t it? Hopefully learning the things they should have gotten years before.

If you call something difficult for a student, that means that the teacher and the student have to try harder. If you call it a problem with abstraction, you are defining it as something the teacher and student have little control over. I call that a cop-out.

24. Steve LaBonne says:

It is indeed a copout. Addition and subtraction involving negative numbers are easily explained in very concrete terms by using the number line. (Multiplication and especially division are admittedly trickier to explain in simple concrete terms.) Premature calculator use is likely to abort this process of understanding- once students have learned to rely on the crutch it will be very hard to get them to take a step back and try to understand what they’re doing.

This is why numerate parents so often have to do a lot of the math instruction themselves and often do a much better job of it than the schools (been there, done that…)- they don’t have a lot of ed-school BS getting in the way of their teaching. I feel sorry for the kids who don’t have that resource.

25. Mike in Texas says:

Steve wrote:

.. .teachers and educrats, who have swallowed the “higher-order thinking” Kool-Aid, like to blame “high-stakes testing” for making them spend too much time on- wait for it- the basics!

Actually Steve most teachers feel the basics are being squeezed out for the sake of high stakes testing, which in Texas is now based on higher order thinking skills, even for 3rd graders.

I would be real interested to see what Bloom herself thinks about how her taxonomy is being applied. I suspect she would say the elementary grades should be the base for the higher order thinking skills in the upper grades.

26. Steve LaBonne says:

Sounds like you may have crappy, fuzzy-minded tests in Texas, and if so you have my sincere sympathy and I share your upset. Testing is only useful when the tests are well-designed. Even though we worship local control and it’ll never happen, a generously funded, well-designed national testing program would be the way to do it right, seems to me. Arithmetic is the same in Texas and Ohio and so should be the standards.

27. Richard Brandshaft says:

“…roughly half the population consists of emotional reaction, associative recall and pattern recognition, while the other half relies more on logical deduction and cause-and-effect.”

In the real world, you need both. I have applied math to the real world. You need
“associative recall and pattern recognition” — sometimes called intuition — to see the math text problem lurking in the real world problem. I’ve seen techies — including me — screw that up.

If you recognize the pattern but can’t quite do the problem, you can look it up or get help. But if you don’t at least partly understand what to do with the math problem, you aren’t likely to recognize the pattern. A lot of this discussion is about which leg of the “two legged stool” is most important.

” How has technology made math change? This is one I’m dying to have explained to me.” Not at the basic algebra level this discussion is about. But as you get higher:

1) We can now do problems that would have been intractable before. This can require methods that were useless before, because they would have taken too much time to use.

2) Should you spend a week trying to simplify the problem to avoid doing a few million arithmetic operations? Not any more. A lot of approximation methods are no longer necessary.

3) Some mechanical steps can be transferred to a computer without loosing understanding. There is no longer a need to learn to use trig table and log tables. You can understand trig functions and logs without that.

28. Jill, your student may be suffering not from a lack of mathematical knowledge or abstraction, but from intellectual laziness. Students who have not been held accountable to think a problem through or fail, develop this laziness. Years of conditioning have taught them that if they just wait long enough or say “I don’t know” enough times, the answer will be given to them by the teacher and they won’t have to work for it. It is a power struggle between student and teacher, and the students learn that the teacher flinches first.

I can get up in front of a classroom of college freshmen and write “2+2= ___” on the board and ask who knows the answer, and not a single student will raise their hand or shout out “4”. It’s not because they don’t know the answer. It’s because no one wants to be the first to break the class’ solidarity and play along with the teacher.