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I’ve heard a lot of the following kind of comment when I discuss curricular issues with people, whether in real life or in the blogosphere: “Sure, we’d love to teach [insert math topic here] to these kids, but they’re just not ready for that.” It came up in the comments on this post about putting geometry before algebra I in the K-12 curriculum (“Students aren’t ready to do abstract reasoning when they’re in the 7th grade”), and it’s come up in discussions I’ve had with colleagues about our freshman math offerings (we were discussing putting some treatment of Polya’s problem-solving heuristic in Calculus and giving students more difficult problems to solve, and the objection was that freshmen “just aren’t ready” for that).
I’m aware that there are psychological theories that establish how children attain different cognitive levels at different times of their lives, so there could be some basis for this idea. But a lot of the time when the “they’re not ready” argument comes up when talking about teaching, it’s just sounds like low expectations and a desire to rationalize the student-faculty non-agression pact. What do you think? Is “they’re not ready” is a valid curricular design principle or just a cop-out?
Update: Alright, so the last question above is loaded. Let’s try it a little more fairly: When is the “they’re not ready” approach valid, and when is this just a cop-out? For example, obviously students who haven’t had algebra I aren’t ready to learn algebra II. But if someone says, for example, that freshmen aren’t ready for proofs, is that psychology talking or is it just low expectations? I’ve heard that last example in the form of “freshmen aren’t emotionally ready” to handle lots of difficult problems, and I strongly suspect that that’s not based on sound cognitive psychology. But I could be wrong.
Casting Out Nines 
In the context of moving from middle school to high sch0ol, there is usually a jump in expectations. Certainly that is the case in my high school.
I have found, in general, that a jump in the level of abstraction at the same time is problematic. We haven’t conquered work habits, we haven’t dealt with kids who are used to ‘getting’ everything facing some frustration.
This is an argument for geometry with substantial proof sophomore rather than freshman year. If you see this as an example of low expectations then we will simply continue to disagree.
Jonathan
I don’t think that, in the case you’re referring to, it’s necessarily low expectations. 7th-8th graders are still young enough that I think the question of transitioning to geometry is valid, although I still think that those students can and should make the transition at that stage. But we could all use some research (is there a psychologist in the house?) to shed some light on it.
I’m more concerned with the cases where this sort of thing is done with college students, like the calculus situation I gave.
Who knows? Decades ago as a high school freshman I took the first year of “Illinois Math,” an experimental “new math” program. First year involved a lot of proofs. For the first time in my life I enjoyed math. At the same time, it was very frustrating for many other students. Good for some, not good for some.
Thanks to the mind set that created No Child Left Behind (and it existed long before the legislation), many teachers have come to believe that they’re not doing their job unless the curriculum is made so foolproof that no thinking is required. Conventional wisdom says that if you don’t break the concepts down so far that the students can just follow steps & get a “right answer” (or if they can’t exactly find a right answer in their notes) then you aren’t teaching. If you leave anything for them to have to figure out on their own, then you’re just teaching to the top level kids.
Naturally, this leads to an aversion to word problems, proofs, & discussion questions. It only allows for very cut & dried, objective types of assessment. It’s also very frustrating for the students. If we would stop trying to cram things down their throats and let them learn to think, I really think we would find that they can.
Of course, as Jonathan said, that’s going to lead to frustration sometimes. I think a certain amount of that would be good for them – the fact that they are never allowed to be frustrated is part of the reason they get to college and say “I never had to study before!” (as you asked in an earlier post)
I really believe that it is the attempt to remove all thinking from mathematics that makes it feel “hard” to so many students.
Having taught elementary kids for fifteen years, I would like to relate one example of how kids might not be ready for certain math concepts.
My father taught me how to tell time when I was four years old, and bought me my own watch, because he was tired of me asking him what time it was. I didn’t have any trouble learning it. So I was very surprised when my own daughter couldn’t catch on to it at six, at seven at eight, at nine…(she finally caught on in Grade 5).
Meanwhile, I taught kindergarten (five-year-olds) for three years. Some of them could learn it.
Then I moved to grade three, where I’ve been for twelve years. They do teach time to the kids in Grades 1 and 2; yet half of them come up to Grade 3 not understanding it. I have seen the Grade 2 teacher spend four weeks on time, and still many don’t understand. I, myself, have made up all sorts of extra worksheets and spent three weeks just on time (when our math book actually allots only two days); yet those SAME kids go up to Grade 4 not understanding. Those same kids don’t get it in Grade 4 either.
Yet, in Grade 5, they ALL suddenly “get it.” After years and years of watching this happen, I have decided it’s a brain maturity problem. So now I don’t worry about it any more. I don’t spend much time teaching time any more, and figure they will get it when they are ready. I try to put the extra time into multiplication and division concepts, which I think are much more important for going up to Grade 4.
What do you think?
I’m sure it’s the same problem for teachers at every level. They get a classroom of students, half of whom can get it, and half of whom aren’t ready, and may not be ready to get it for even several years after that. So what do you do?
Eileen
Dedicated Elementary Teacher Overseas (in the Middle East)
elementaryteacher.wordpress.com
I say its always a cop out… That is just the teacher’s way of saying “I’m not ready to take on the responsibility of trying that hard.” Or its a parent saying the same thing.
Yah, that’s probably harsh, but I am so sick of our intelligence as human beings being so disgraceful. We have so much potential that is not being utilized. As for the psychology excuse, there are ways to help people reach that next level of understanding, but no one seems to care.
I know that I am not a teacher, and that I have no experience in a classroom, but I will stand strong on this opinion. I have never failed at teaching some one else a concept that I fully understand. It just takes a bit of time and a lot of patience. I think that second thing is what people have so much trouble with.
I read your post with interest because this is an issue that I really struggle with as well. To some extent, I wonder how much of it is a self-fulfilling prophecy? If you are pretty sure someone can’t get a certain problem, what is the likelihood they will? After several years of posing the problem and having no-one get anywhere, why continue? I take great joy in posing really open questions that let students explore and find their own way. I think it really makes students uncomfortable, but once they get into it, they can be extremely impressive (often, they do far more work than I ever would have dreamed of asking for with a more narrow closed assignment).
I also think there is another side to this. It takes two to have a good learning experience, both the instructor and the student. I worry sometimes that we are doing a great job of shutting down student creativity and curiosity and leaving them with a “I don’t care, what’s the answer?” attitude. Once this attitude is ingrained, it seems extremely difficult to overcome. Even a few students with this attitude can drag others down in a classroom…
I think Eileen makes a good point about development, and time makes a good example, since the concept of time exists in that fuzzy area between concrete thinking and abstract thinking.
If we take Piaget’s Theory of Development as a rough guide (yes, it’s old and unrefined, but it IS straightforward), age 12 is roughly when children first develop the ability to deal with abstract concepts. Note that this is right around sixth grade, which might explain why a transitional idea like time frequently clicks right around fifth grade. This also suggests that logical concepts could be introduced as early as sixth grade. Anecdotally, I had exactly that. I was in an advanced mathematics program in middle school. In sixth grade I took a course in so-called “Number Theory”, although it was actually rudimentary concepts of Set Theory and Abstract Algebra (sets, concepts of one-one functions, modular arithmetic, ciphers, other fun stuff), and the second semester we also did Symbolic Logic, where I was first introduced to something like a two-column proof. Seventh was Algebra I, and Eighth was Geometry. So, it’s possible to do this, however it’s clearly not a “general curriculum”. Not even close. That said, I think rigorous logical thinking can, at least, be introduced earlier.
Certainly, I think Logic ought to be a subject. There seems to be a lot of flailing about in middle school mathematics that I’ve never really understood. There’s this focus on odd data analysis and methods mixed into pre-algebra that I’ve never really understood. I would almost like to advocate taking a semester and stepping back from numerical mathematics to introduce some of the more thinking aspects. All of math becomes easier once the concept of logical thinking is solidified, and I think this can happen earlier if we put logic earlier in the curriculum, like before the second year of math/philosophy-major university education!
I was thinking about this earlier with the post on the Geometry before Algebra I, and I thought, why not remove the algebraic content in Geometry, and replace it with some degree of Logic. My own eighth-grade geometry class used propositional logic to demonstrate how to set-up a proof. Proofs, regardless of style, all have the same logical base; I think if we taught this earlier, kids would not only find Geometry easier and do-able before Algebra I, but would develop the sort of thinking skills that apply to Algebra, and perhaps even more importantly, to other subjects: History, Literature and Science courses all require logic to argue a point, and Middle/High School Mathematics should, in my opinion, exist to create citizens who are not just numerically literate, but also capable of understanding a logical argument. Math doesn’t just teach this, mathematics IS logical arguments. Math is more than addition and division algorithms. It needs to be taught as such. Besides, Logic and some of the less numerical stuff is not only less intimidating (you get pictures and words!) but it can also be a lot of fun.
So I do think it’s a bit of a cop-out. I think Geometry could be taught before Algebra. Although really, I think Geometry, Logic and Algebra could all be rolled together into a sort of three-year course. I know I’ve seen others suggesting this, but it makes sense historically, and as a student, I think it makes sense from a pedagogical stand-point too. Elementary Algebra and Geometry are really not separate disciplines. We already teach basic set theory to elementary school students (Venn Diagrams, anyone? I remember them from something like Third Grade, and then recently in a University math course).
Math isn’t hard. Thinking is hard.
In sum: Standardized tests are still ruining everything.
James
James: The geometry book I use in my college junior/senior course does a nice job of blending in elements of logic and proof right along with the geometry content. I’ve often thought that book would go over well at the high school level.
Robert,
For the schools that can afford the software integration (another issue in itself), I think a text with such a focus that brings in technology, and perhaps most importantly, a feeling of experimentation, would go a long way.
I think we’ve dropped sets from most elementary curricula. Venn diagrams too.
Again, some may think that delaying geometry is a cop out, but when an alternate is proposed (in this case, one year later), it should give pause.
I agree that logic should be taught with geometry. Some elements are part of most standard texts in the US.
And then for us, logic is also a one-term senior elective. I like teaching that.
Jonathan
I am working right now with my daughter, trying to teach her a lot of things which she hasn’t understood up to this point (which were taught in Arabic), as she is now switching to an American school, where concepts will be taught in English.
I am using three different books. The best book by far is an English text from the 1880’s. I’m also using a pre-Algebra grade 7-8 book from America from the late 60’s-early-70’s. I went through school in the 1960’s and this American book is talking all about sets. I see now what confused ME as a kid was all this talk about sets. I am SO glad to hear they are dropping all this out of elementary curriculums (not that I’ve been teaching it).
The old English book I’ve been using was written by a master at Eaton, and while it has no algebra, it explains things in SO much more detail than the modern books, AND has SO many more examples ALL with ANSWERS in the back (I’d estimate it to be a Grade 8 book).
I think many classes today are skimming over concepts, short on number of problems kids have to do. Then the kids get their books back where problems are marked right or wrong, BUT THE TEACHERS DON’T GO OVER THOSE PROBLEMS, so the kids don’t LEARN from their MISTAKES.
In my class of eight-year-olds, I give my students an A or an F just for having all their homework done, which I do not actually record in their grades, but the students THINK it is recorded in their grade. That gets them ALL COMING TO CLASS WITH THEIR HOMEWORK DONE, and READY TO LEARN. I have them put away pencils and erasers, and get out their own ink pens to check their own work.
First, I go around the room and mark a large A+ in red on the people who have their homework done (doesn’t matter if it’s right). For people who either didn’t DO their homework, or didn’t FINISH it OR who DIDN’T SHOW THEIR WORK, I give them a very BIG F in red, AND attach a “homework alert” paper, which their parents have to sign (doesn’t matter if they have correct answers). Of course, I have sent home a paper (also glued into their homework books explaining all this, so the parents know it, but the students themselves have never bothered to read it).
Then as I give students the correct answers, they either put a check mark indicating they have the correct answer, or cross out neatly and mark the correct answer, if they missed it. (This also makes it easy for parents to see exactly WHAT the child needs to study before the chapter test.) Because they ALREADY have an A+ (even for wrong answers) it cuts down on erasing, and changing of answers, just for the benefit of the grade.
All of this is not my own idea. After teaching Grade 3 overseas for a year (and having a secondary, but not primary teaching certification) , I went back home to Colorado, and spent a week observing third grade in my own childhood school. I found the teachers teaching math by the method I described. I came back and immediately adopted these practices.
We then go over problems on the board. I feel so great, when kids (because they show their work) can say out loud, “Oh, here’s what I did!” And by so doing, they learn from their mistakes.
I’ve had really good results this way.
Eileen
Dedicated Elementary Teacher Overseas (in the Middle East)
elementaryteacher.wordpress.com
Eileen: What’s the title, author, publisher of that book from the 1880’s you’re using?
It’s “Arithmetic for Schools,” by Rev. J. B. Lock, M.A. (Fellow and Bursar of Gonville and Caius College, Cambridge, Formerly Master at Eaton). New Edition, Revised and Enlarged. London, Macmillan & Co., Ltd. (First edition 1886. Several subsequent editions, this book’ latest edition 1897.) I purchased this book in an antique store in Denver, which imported things from England. I never imagined I’d be using it with my own daughter 30 years later!
I find it interesting the way this book is organized, basically with the same types of math topics we cover today, but far less THEORY and far more PRACTICAL knowledge. Theory IS included, such a a complete description of the Sieve of Eratosthenes (which I had never heard of before reading this book).
I think one of the biggest problems in math education today (and have really been reinforced in this opinion by the old book I’m currently using) is that students are not MASTERING certain topics before moving on to the next. Instead, they are given a “smattering” of a lot of areas, not really mastering many of the skills, which continue to be treated (unsuccessfully with many students) in future math years.
I’m currently working on factoring and fractions with my daughter. There is a method of factoring into primes which is taught in this book which I had never seen, but that I really like, even though it took me time to work out.
In the book I’m using from the 1965-1970 (Pre-Algebra Mathematics, by Eugene D. Nichols, Publ. by Hold, Rinehart and Winston, Inc.), here is an example of the typical addition of fractions problem: 2/3 + 5/6 + 2/9.
Here is a TYPICAL example from the old English book: 1/17 + 32/34 + 3/8 + 3/56.
Or, another: (add together and simplify):
21/8 + 22/9 + 23/10 + 24/11 + 25/12.
You should SEE the story problems this book gives. (I haven’t even tried to do these myself!) Here are two examples:
1.) A father divided a piece of land among his three sons thus; he have 12 1/4 acres to the first 3/8 of the whole to the second, and to the third as much as to the other tow together; how many acres did the third get?
2.) A man gives away to each of our people 1/12, 3/35, 3/28, 2/21 of a basket of apples and has only just enough apples to be able to do this without dividing an apple; how many apples had he?
Now compare this with the math ability of students today, and with the much simpler story problems in books today!
So what do you think about this?
Eileen
elementaryteacher.wordpress.com
Thank!!! It good content. It good blog for sharing.
I enjoyed this post and the comments very much.
I agree with Eileen completely, and have come to the same conclusions as an educator. Waiting for ‘readiness’ isn’t about certain math concepts not being presented to students, but that many times mastery should not be expected for every child at the same age.
It grieves me when kids are told right out the gate (with below average grades) that they can’t ‘get it’, when in reality they weren’t developmentally ready to understand and master the material. This sets them up for failure, and is the opposite of nurturing.
Anecdotal comment is all I have, but for what it’s worth….
When I was in 7th grade, which was the beginning of junior high school for me, I was in an honors math class (middle schools didn’t exist yet). The teacher gave us an 8th grade math book, and told us we had to leave it with her, and couldn’t take it out of the classroom. Sometime during that year, she introduced us to Algebra, and I was totally lost.
I have a good brain, and an above-average IQ, but I could not understand how one could add letters together. The teacher would ask me: “what’s the answer to a+b?” I had no clue, and would suggest that maybe it was “c”? And she would ding me for being a smart-alek.
I limited myself after that, only taking general math courses until my counselor told me I had to have some higher math courses if I wanted to go to college. We had moved by then, and in 10th grade I took Algebra I. That teacher had the ability to explain things in plain language without getting impatient, and my brain was apparently ready for the abstract concepts by then, because I got As in her class. And I got good grades in the Geometry class I took as well, but I never took anything more challenging.
I spent 2 decades convinced that I couldn’t “get” math, because of that 7th grade teacher, and trying to learn something when my brain most likely wasn’t ready for it..
elementaryeducator: I got 50 acres and 420 apples by examination (no pencil involved). Do I win a cookie?
Thinking may be hard, but it gets easier with practice. Maybe that’s what we should be demanding of students: reason their way through things, and add in bits of irrelevant material to the advanced exercises to make them reflect the real world.
I have taught high school math for 18 years and can tell you beyond a shadow of a doubt that some kids are ready for Algebra way before other kids.
When I was department chair at a small Christian school, we would administer an “Algebra Prognosis Test” to incoming 7th graders to determine which students would be on the honors math track. It would introduce Algebra concepts and then ask students questions about the concepts. It basically distinguished between which kids could understand the complexities of abstract thinking and which could not. In contrast, I taught a low level 9th grade Algebra class a few years ago in a public school, and although the class was full of many students who could care less about math (let alone school), I had a handful of kids who really wanted to learn Algebra, but JUST COULDN’T GET IT. They weren’t ready for the abstract thinking.
As far as Geometry before Algebra, here in Texas, 8th grade math has a large component of geometry concepts, taught before all kids have to take Algebra I in 9th Grade. All kids then take Geometry in 10th grade. Sophomore Geometry covers more in depth the concepts introduced in 8th grade (as well as new concepts), just as Algebra II covers more concepts than Algebra I.
The reason that I believe the textbooks have changed so much is because as few as 25 years ago, not every high school student took Algebra I – only the students who the teachers felt were ready for it. The older books could be meatier because only the kids they knew were ready for the material were studying it. Now all students have to take Algebra, and we as educators have to teach those who are ready as well as those who are not. The texts are trying to help those kids who aren’t ready for Algebra, but all they are really doing is watering it down for the kids who are ready.