It was 1977 and I was taking a child development course for my Master’s level program to become a special education teacher. The assignment was simple. Take two children of different ages and conduct some learning experiments on them. The objective was to see if Piaget’s theories were true. I needed two children, so I asked my cousin if I could borrow her two kids—David and Rachel (who still remember as adults the fun we had doing all this). David was eight-years-old at the time while Rachel just turned five. Of course, I did that famous conservation experiment of pouring water into a tall thin glass and the same amount of water into a wide narrow glass. Obviously, both thought that the tall narrow glass held more water even after both watched me pour the same amount of water into both glasses. What is the importance of this experiment? Well, young kids think very concretely and not abstractly. And how do you develop abstract thinking skills? Give them many of concrete experiences over a long period until their brains become mature enough to understand the abstract concept that no matter what the shape a container may be, if you pour the same amount of liquid into that container, it is still the same amount
This experience I had with my cousins came hauntingly back to me about a month ago when I tried to teach a bunch of fifth graders to estimate fractional sums using benchmarks. The concept appears simple. Take a number line, start at zero, make several benchmark points, such as ½ and 1, and then estimate whether a given fraction is close to these benchmarks. For example, if we add 7/8 and 3/8, we should estimate that our answer will be about 1 and ½. Obviously, 7/8 is close to one and 3/8 is close to ½. With our adult minds, this is a no brainer, but not to the fifth grade mind—especially a mind that may have a learning disability. My kids just did not get it. I used every special education, multisensory method on the books. I color coded, used fraction bars, as well as visual illustrations, etc., etc. Most just wanted to add the like denominators and did not want to estimate first using this method. I then decided to question them intently to understand why they were having such difficulty estimating. To my amazement I discovered the reason. All my concrete manipulatives and illustrations confused them even more. Each manipulative and drawing was a different size and they did not understand that no matter the size or type of fractional illustration presented that the fraction was really the same size. It was hard for these fifth graders to understand that if I cut a pizza in or a jelly bean in half, it was still a half. Therefore, I made a fatal teaching error that many new teachers make. I assumed knowledge or understanding that my students really did not have. To prove their lack of understanding, I took two jars of different sizes and poured a glass of water in each. I asked them to write on their personal white boards which cup had more water and most chose the tall, thin cup again. Piaget came hauntingly back and now I understood why these LD kids were having such difficulty with this common core concept. I realized that it was not that many wouldn’t learn it, but that many couldn’t learn it. They were just not ready.
Furthermore, I tutor several middle school students in math. I work with one learning disabled 8th grader who is, with a lot of extra help, passing within an integrated setting. Fortunately, his parents have the resources to purchase my services for three hours a week. In addition, not only does the student have a highly experienced special education co-teacher in his math class, but he also gets additional special education teacher support services three times a week within a very wealthy suburban school district on Long Island. To his benefit, the student although learning disabled has strong intellectual potential that enables him to easily learn the various strategies I and his teachers have developed to help him do the math. Yet, when tested on these concepts, he mostly gets grades in the low seventies on tests in which problems contain three or more steps and which requires him to describe using mathematical terms various math processes. One problem he got wrong had to do with the Pythagorean Theorem. Mathematically, he knows the formula and can apply it to solve problems presented algebraically. He understands that if we want to find the unknown length of one side of a right triangle, he can do so as long as he knows the length of a hypotenuse and an adjacent side. However, on a test in which a problem derived from a sample CCLS standard, he got completely lost. The problem had a right triangle containing adjacent squares for each side. The question asked what assumption the student can make about the area of the largest square. Furthermore, he was expected to explain his assumption in mathematical terms.
After looking at the problem, it appeared familiar to me. I then remembered where I saw a similar problem. I decided to take a trip to my attic and opened up an old box. Within the box, I found my high school review books. After a little skimming, I found a very similar model problem—within my 10th grade Amsco geometry review text. Then I remembered the difficulty I had with my first term of geometry in high school and all the extra help I needed to master and understand those theorems at the time. Now we expect a student to master concepts that used to be taught to 15-16 year old students thirty of so years ago. A 16 year old student is well into what Piaget calls the formal operational stage of development. Those are fancy words that mean that a student of that age can more easily understand very abstract concepts. Now we are supposed to expect a 13 year-old student to have the same capacity as a student that is very close to college age. Obviously, some 13 year-old students can understand such concepts, but most will have difficulty, again, because they may not be developmentally ready—especially if a disability is present. When I recently stated this at a meeting, I was told that I have low expectations for students. I replied that I do not have low expectations, but realistic expectations. And that these expectations are based on a good deal of scientific research.
The Common Core curriculum appears to be one that was developed by anecdote and not by research. I remember when my youngest son graduated from high school, the Valedictorian was an Asian young man who came to the United States two years previously without knowing a word of English. I recall the Principal saying to the audience that it was possible to accomplish so much when one perseveres and works hard. What he didn’t mention was that this student probably had an IQ that was through the roof! It would be unreasonable to expect other immigrant children to accomplish what this student did when research has shown it takes an older student five to seven years to learn enough academic vocabulary to perform well in an English language school. One should not build a curriculum that could only be easily mastered by above average and superior students that make up only 15% of the total population.
Interestingly, just yesterday I received an email from my school district which contained a list of math vocabulary terms students are expected to master at each grade level. When I looked at the kindergarten math vocabulary, there was the term “decompose” which means to break down complex numbers to get a better understanding of place value. To expect a kindergarten student to understand and use this concept is beyond ridiculous. When I was in Kindergarten, I am pretty sure I had no idea what this term meant and I am also sure my kindergarten teacher had no interest in teaching me its meaning when her greater concern was that I know how to write my name, address and phone number in case I got lost. I really don’t think there is any necessity for a five-year-old to use college level vocabulary to explain complex math terms when many still need to develop one-to-one correspondence. Of course, someone who supports common core would say that all they are doing is raising the bar. However, this is a bar that is twenty feet up and for a five year old impossible to master. By the way, yesterday I asked three kindergarten students to decompose the number 12 and they replied with blank stares. I have been involved with educational testing for nearly thirty years. A good part of my career involved administering diagnostic tests to determine if students had learning disabilities. I clearly remember when I was taking courses in diagnostic assessment, a professor saying to us that when most students fail a test, the problem is not with the student, but with the test. Therefore, if most students at a certain age will not be able to master these so-called common core standards, the problem is not with the kids, but with the standards. Standards that unfortunately violates every rule of child development.
Great article supported by facts. Irrefutable logic, but I suspect common core is happening anyway. Thanks for debunking the idea that the CC was developed by people who understand research and learning theory. There are several things about the common core I like, but the idea of “building the airplane in the air” which means experimenting on our children is not one of them. What do we know, we are just professional educators.I guess this is a matter best left to legislators.
There are many things in education that can be improved, but we should first do no harm. The changes must be based on sound research, not a knee jerk reaction of a skewed international statistic.
“For example, if we add 7/8 and 3/8, we should estimate that our answer will be about 1 and ½.”
This is not what the standard 5.NF.2 requires, and in fact is harder than what that standard requires. It asks that students use benchmark fractions to assess the reasonableness of an answer. So you provide a sum of two fractions and an answer then ask if that answer is reasonable and why it is or is not. In your case you’re providing two fractions and asking them to round both of them then provide a rounded answer. Also, you said your kids just wanted to add like denominators – there’s your first problem, they don’t even know the fourth grade standard of adding two fractions with like numerators.
If you look at the example provided by the standard, you are supposed to provide a sum of fractions that do not have the same denominator – otherwise why wouldn’t they just use their fourth grade knowledge and say that 7/8 + 3/8 = 10/8? You’ve given them a false thing to do (use estimation and rounding when it isn’t necessary).
How about this question, is 1/8 + 2/3 = 3/11? 2/3 is bigger than a half and 3/11 is less than a half. Thus 3/11 is not a reasonable answer for the sum of 1/8 and 2/3. That is the question that follows the standard.
At this point you can continue with adding the fractions using a common denominator, but that is a completely different task and set of steps. You might want to read this document by Hung Hsi-Wu (one of the members of the CCSS board, faculty emeritus of math a Berkeley) which discusses how to teach fractions from grades K-8: http://math.berkeley.edu/~wu/CCSS-Fractions.pdf
You do expect a mixed group of fifth graders to manage your question, below? To me, that violates the principles established by Piaget for pre-operational, concrete operational, formal operational stages of child development.
“How about this question, is 1/8 + 2/3 = 3/11? 2/3 is bigger than a half and 3/11 is less than a half. Thus 3/11 is not a reasonable answer for the sum of 1/8 and 2/3. That is the question that follows the standard.”
First paragraph, last word, replace numerator with denominator. Thanks.
Children in Kindergarten do not need to know what the words compose and decompose mean as they are written in the CCSS. They must learn that 2 and 3 make 5. They must learn that 10 and 10 make 20. The language of the CCSS is problematic and the US Dept. of Education would do well to provide videos of what the learning is, in examples of each standard. The K standards are not very different from the NY state standards already in place, when you do the translations for language of each.
And now some states may try to base family eligibility for welfare on their child’s test scores!!
There are lots of problems with the Common Core. Two come to mind.
First is that they are linked to standardized tests and there are powerful education companies that have a lot of money riding on these tests becoming the real Core — a money making dynamo. That will result in teaching to the test and standardized instruction that will be do even more damage to creativity and critical thinking.
Second, standards should be voluntary — suggestions, not edicts.
Why? Both to allow them to develop naturally as the culmination of an interactive process
and to trust in the professional judgment of teachers
The best explanation I can come up with for the latter is quoting
Maja Wilson’s ‘kind of like an artist’ article about the common core http://umaine.edu/edhd/files/2010/08/Wilson-Kappan-article.pdf :
“This problem can’t be solved with more, bigger, better programs.
It must be solved from the inside by, first, acknowledging that
teachers’ decision-making process is at the heart of good teaching,
then by allowing teachers to actually make decisions, and
finally, by reflecting on and evaluating those decisions in light of principles.”
Great post and great comment by Brian Ford. I had a short stint as a math teacher and tutor ten years ago and noticed that many math problems posed by the middle school Massachusetts MCAS tests seemed to be testing IQ and reading ability (the reading ability is also influenced by proficiency in English) . I also observed that the science teachers were expected to do a big unit on hypothesis testing for sixth graders. As you point out, Piaget’s theory would predict that many sixth graders do not yet have the brain maturity to handle abstract reasoning, but in my experience it is not entirely a matter of maturity. I have noticed that there is persistence insofar as some people (a third?) are always able to handle certain abstract ideas easily and for others (another third?) it never becomes easy. I wonder if it is related to the ability to think several moves ahead when playing chess. I can do this because I have a sort of mental workspace in my head where I can manipulate things and see how they look. I don’t think everyone has this. I know that when I was a teaching assistant for a university course called Statistics for Business Administration, about half the students had a very hard time with the concept of hypothesis testing. To be comfortable with this topic, it helps to have the mental space to hold multiple possibilities at once.
I once led a few family math workshops at a public library near a university. A father who was a PhD student in Computer Science came with his two sons. I got them started on a math game that is similar to Battleship except it uses graph paper and graph coordinates and in addition to announcing a hit or miss one must announce how far away the target is, as measured in “city blocks.” The boys really enjoyed this game, but after a while we ran out of graph paper. To my amazement, the boys, (maybe 7 and 9 years old?) continued to play the game by envisioning the graph in their imaginations!
During time I spent as a substitute teacher I discovered that many special ed teachers and assistants fed answers to middle and high school students for routine tests and homework because they wanted to protect these students who were mainstreamed from the constant sense of failure that they otherwise experienced. Now as all students are tested on topics, like hypothesis testing, that test an ability that is persistent and varies widely, those without the ability are reminded year after year (or if there school is under performing, far more frequently) that they are stupid!
When I say that this ability is persistent, I suspect that it either comes from genetics, or from stimulating experiences (music?) occurring in a safe and nurturing environment in the first 4 or 5 years of life. So, if you do have a large mental space to manipulate objects, it really helps for very specific types of problems. If students don’t have that space, or it is severely limited, the student must make due with other strategies. I am supposing that a student’s access to a mental space to manipulate objects doesn’t increase over time. Do others agree?
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