BA/MS in math
MA in special ed
EdD in math education [ABD]
15 years teaching math from middle school to college and two serarate times as a math coach
National-board certified in Adolescent/Yound Adult Mathematics
I think that more than education in math, what we need are teachers that actually know progressive pedagogy. One of my son’s math teachers claimed he used constructivism. I asked him what constructivism is and he said, “A way of creating knowledge.” When I pressed what that meant, he told me I should read about it so that I knew what I was talking about. He didn’t know that while going for my EdD at Exeter I studied constructivism in math ed and it was clear he had no clue what constructivists practices were. It is shocking to me that most teachers really do not understand what students need in order to learn.
But that’s not really the problem. Our educational system is 120 years old and was created so men could run a farm or small business and women to be a farm-wife with her egg money or be a secretary in the city. Teachers went to normal schools to learn how to teach, often without a degree in a non-educational field and as soon as she got married she had to leave the field so good luck getting real-life experience in the classroom. Our educational system is still at heart the one started by John Dewey in the early 1900s. The comparison I make with all of these new programs and inititives is take a 120-year old how and fix the roof and repaint it but don’t fix the walls or the foundation. What do you have? A 120-year old house that looks nice until it falls down. It wasn’t until NCLB that it was actually required that teachers be educated in the field they teach (some states had the requirement earlier so YMMV).
Elementary school today still reflects the one-room schoolhouse. Why do we expect an elementary teacher to be well-versed in subject and pedagogy x 4? How much math, science, English and social studies do we expect them to know and be adept at teaching? It’s not fair to them and yet it was only recently that we thought that maybe elementary teaching duties should be split between math/science and English/social studies experts. Now add in the lack of mathematical knowledge in elementary teachers along with the fact that math is an anathema in many elementary classrooms.
So my solution would be to completely change the structure of education to represent the 21st century. We could make elementary/middle school modular rather than the one-size-fits-all of a fouth-grade class. We should measure students with a growth model so that if I get them at a 6.2 GLE I know exactly where they’re at and if they leave my class at a 7.1 GLE you know how effective I was. Real-life skills should be emphasized. Teach students to write a resume or an email (or SD post)that won’t be misinterpreted. Teach students how to run a small business. Teach students how to look at how events in the past created the world today e.g. w hy illegal immigrants are protected as “residents” despite not having lawful presence. There is a lot that can be done beyond 5 paragraph essays, quadratic formula, the periodic table and memorizing the names of the first 7 presidents.
There was one outside the Seattle’s Pacific Science Center that I remember from 4th grade. It had to do with bouncing balls and the normal curve but what still facinates me is how every 10th(?) ball was black and would end up in two and only two columns.
that is my favorite too. as a youth when I went to a museum with one i would always watch a few cycles to see what happened. the one i saw had only one color ball.
the device is called a Galton Box. making one is on my bucket list.
Does it matter very much that my son (13) still hasn’t quite got the hang of multiplication tables? Despite the flash cards, and memorization drills, and having him write the tables over and over… he still can’t remember a third to half the facts at any given moment.
Does it mean he’s doomed to stuggle in math through high school and the rest of his life?
Teachers emphasize the multiplication tables because in our days we didn’t have calculators with us all of the time. Now with cell phones everyone does. So is there a difference in estimationg a 15% tip that I roung the number, take 10%, halve it and add them together (e.g. 27.41 => 30 => 3 +1.5 => $4.50 tip) while Mrs Cad pulls out her phone and uses the tip calculation app? I had this exact same discussion with a teacher who complained his high-school students didn’t know their times tables. I asked if the students were allowed to use calculators and he said yes so I asked then why do they need to learn their times tables. He had no answer.
Will learning his multiplication tables help him be fluent? Yes but fluency while nice should not be something that means the difference between pass and fail. If so, then you have a bad math teacher.* There are other things he should memorize too to make his life easier in math classes. He should know all of the prime numbers 101 and below. He should know his first 20 perfect squares. Those are very useful in math and not available on a calculator.
*Personal note: I had to fight for my son in 4th grade because he had not “mastered” multiplication and they were going to hold him back. They were measuring him by time so out of 100 lets say he did 65 problems and got 2 wrong he got 63/100 thus failing the assessment. I argued that considering he met the “mastery” standard on the ones he did complete and if untimed he could do all 100 and meet the “mastery” standard that what they were measuring was fluency and not mastery. Since fluency is not a requisite for moving into 5th grade they let him move on. But the point remains that an elemetary teacher and administrator both confused speed at calculating with knowing how to do the calculation and had I not been a math teacher myself and could speak with authority on the subject, could you imagine what would have happened?
** Stories like the ones above and my whole experience with the latest RtI process (great story if you’re familiar with special education or Section 504s) for my son is why I’m his biggest advocate and am that parent that marches down to the school. It’s not that I think my son is perfect, it just that I have seen so many teachers and administrators that don’t know what the fuck they’re doing.
Honestly, the best strategy is the simplest. Make sure she has plenty of time to do the work. Also, I would ask for more specifics on her processing delays because typically the weakness is in one area. If for example she has difficulty with visual processing, it would help to read the question or directions to her orally.
As I’ve said before, memorization is not as critical as it used to be when it is so easy to look things up. Your daughter is nine? By high school she should be able to be googling “quadratic formula” on the smart phone/pad that will exist by then. The probem is that you have a lot of old-school teachers that will require memorization because that’s how they learned, their pappy learned and their grand-pappy learned. My suggestion is if you run into that problem have you daughter get a 504 plan that would get her the acommodations she needs.
Thank you
Unfortunately, without getting into limits and fluxions, it is difficult to explain the why and even if we did that it is still mathematical symbol manipulation so I’ll see if another approach answers the “why?”
A tangent is a line that only touches a circle at only one point. This line is unique meaning that there is only one line, the line perpendicular to the radius and intersect right where the radius touches the circle. It is not difficult to expand this definition of tangent to any curve as long as we do so locally meaning in a small area around point of intersection.
But there is another tangent. The (in)famous opposite/adjacent of trig days. Interestingly the exact same diagram as the rise/run or delta Y/delta X diagram from Algebra I defining slope. Because of this, there is a relationship between algebra’s slope and trig’s tangent although the two are rarely connected until you get to calculus. The interesting thing is that geometry tangent and trig tangent are related as well and so in calculus the derivative is the slope of the line that is tangent to a curve at a certain point so the simplistic answer is the derivative of x^2 is 2x since given the point (x, y) the slope of the tangent line is 2x so for example through the point (3,9) the tangent line is y = 6x-9
This comment has me puzzled. The first 20 perfect squares are indeed available on a calculator! You can use any calculator to find 1x1, 2x2, 3x3, etc. Assuming the calculator has a square root button, it can also be used to easily tell whether any given number is a perfect square or not.
Now, it is useful to be able to look at, say, 49, and recognize (without a calculator) that it’s a perfect square (namely, 7 squared), which I’m guessing is the kind of thing you meant. But it’s also useful to be able to look at 42 and recognize that it’s 6x7, which involves knowing multiplication tables. How can you be any good at factoring if you don’t have your multiplication facts memorized?
Indeed. What I meant is that calculators don’t have the list of perfect squares. What you give is a way of deriving that list via calculator but that doesn’t really help when simplifying sqrt(75) or sqrt(80).
To address your second point, using prime factoriztion, it is not terribly difficult to show 42=2x3x7 which would give us our factor pairs. Even without prime factorization they can divide 42 by every number 1 to sqrt(42) to find the pairs. The difference with memorizing primes or perfect squares is this technique can be done on any calculator the student may have on their cell phone and thus does not rely on memorization although I do agree to factor fluently the student should know their tables.