Is This the Key/Secret to Learning Math?

[bballwaterboy]

Does this 81-year-old hold the key to teaching kids how to understand math? (Jan. 24)

https://www.washingtonpost.com/loca...4f6116-c12e-11e5-9443-7074c3645405_story.html

But week after week, Johnson still drives from her Clinton home to the after-school program at the Washington Tennis and Education Foundation campus in Southeast and, according to officials at the program, consistently lifts underperforming math students to grade level and beyond. This month, she received a copyright for a diagnostic test that she says can assess specific gaps in students’ math knowledge in minutes.

She’s now working with the foundation to raise money to digitize the test, which includes eight to 10 math problems for each grade level, so that it can be used in schools throughout the country.

“I believe all students can learn math if they understand the laws,” Johnson said. “If a student masters a problem on my test, I don’t care what test they take, they master it.”

From this week's Washington Post. Any thoughts on this anyone?

Sounds kind of cool, actually. This part does seem very important:

Johnson’s methods rely on drilling in the basic concepts of math — or, as she puts it, “the laws of math” — and ensuring that students understand why each step of solving a problem is necessary.

If a student doesn’t conceptually understand that they can’t add apples and oranges together, for example, how will they know what to do a few years later when they see different variables in an addition equation in algebra class.

 

[symbolipoint]

Public school administrators and most teachers will tell you that you are crazy, if you tell them that studying Algebra(Basic Algebra, like in "Algebra 1") will teach students the rules by which numbers work. The teachers and administrators will tell you that Algebra 1 is too advanced, and that students must first master Basic Arithmetic before they are ready to learn the rules.

 

[Student100]

The article doesn't really say much of what she does, so it's hard to have any thoughts on it. Mathematics doesn't have "laws".

 

[brainpushups]

It's sad that it seems revolutionary to make sure students understand what they are doing, though I agree that this seems to be rare. I encounter plenty of high school students that have a poor understanding of basic operations.

On a lighter note, this reminds me of a cartoon I saw the other day:

http://www.smbc-comics.com/index.php?id=4000

 

[ZapperZ]

Isn't it rather strange that we are still talking about the "best" way to teach kids mathematics in 2016? Considering that the US is often cited as being behind China, Korea, Singapore, Norway, Sweden, etc. in terms of students' knowledge of mathematics by the time they finish high school, why are we trying to invent the wheel? How come no one is looking at how those countries are teaching their kids in math?

Is it because those kids spend a lot of time with their school work, and kids around here simply won't put that same type of effort? So we're trying to invent a "short cut" and end up messing things up that our kids suck in math when compared to students in those countries?

Zz.

 

[WWGD]

Isn't it rather strange that we are still talking about the "best" way to teach kids mathematics in 2016? Considering that the US is often cited as being behind China, Korea, Singapore, Norway, Sweden, etc. in terms of students' knowledge of mathematics by the time they finish high school, why are we trying to invent the wheel? How come no one is looking at how those countries are teaching their kids in math?

Is it because those kids spend a lot of time with their school work, and kids around here simply won't put that same type of effort? So we're trying to invent a "short cut" and end up messing things up that our kids suck in math when compared to students in those countries?

Zz.

I think it also has to see with the cultures in those countries, which value knowledge over fame, popularity, etc. Though maybe these cultures have the disadvantage of being too tied down to tradition and not having enough people willing to rock the boat ; there are plenty of this type in the U.S.

 

[Mark44]

The article doesn't really say much of what she does, so it's hard to have any thoughts on it. Mathematics doesn't have "laws".

I've seen textbooks that have sections on the "laws of exponents," "associative law of addition," etc. Many books use the term "property" instead, but these two words can be considered synonyms.

 

[Student100]

I've seen textbooks that have sections on the "laws of exponents," "associative law of addition," etc. Many books use the term "property" instead, but these two words can be considered synonyms.

Property is a better word I think, but that's just arguing semantics I guess.

 

[Mister T]

There is no key or secret to understanding math. Teaching strategies that work for one teacher may not work for another because teaching is an interaction between people, and different people have different personalities, attitudes, expectations, etc.

American society needs to focus on empowering teachers and holding students accountable for learning. Currently neither of those things is happening to anywhere near a large enough extent. The problem gets worse as each generation of under-prepared students become the next generations' teachers.

Hopes placed in quick fixes like this are distractions from the much deeper issue that is the real problem.

 

[UncertaintyAjay]

The real key to understanding math is to love it. If you love it, you are going to put in the time and effort to know more, to understand more. Also, everyone learns in different ways so anyone way of teaching math is not going to work for everyone.

 

[WWGD]

Maybe the best you can aim for is to have teachers who are well-prepared and enthusiastic, who can transmit their enthusiasm to students through their teaching.

 

[Mister T]

Maybe the best you can aim for is to have teachers who are well-prepared and enthusiastic, who can transmit their enthusiasm to students through their teaching.

That is certainly an important part of the effort. Unfortunately the way administrators, parents, and students treat teachers, and the way students are not held accountable for learning interfere with that effort. Most teachers have their spirits broken. Or never consider adopting teaching as a profession in the first place because of these issues and the low pay.

 

[WWGD]

That is certainly an important part of the effort. Unfortunately the way administrators, parents, and students treat teachers, and the way students are not held accountable for learning interfere with that effort. Most teachers have their spirits broken. Or never consider adopting teaching as a profession in the first place because of these issues and the low pay.

Yep, " how we appreciate our teachers" is another empty statement along the lines of " our employees are our greatest assets" (to be fired when we want to squeeze $1 in profits, in the short run), "we love our customers" , etc.

 

[1oldman2]

I think it also has to see with the cultures in those countries, which value knowledge over fame, popularity, etc. Though maybe these cultures have the disadvantage of being too tied down to tradition and not having enough people willing to rock the boat ; there are plenty of this type in the U.S.

I suspect you may have hit the metamorphic nail on the head.

 

[WWGD]

I suspect you may have hit the metamorphic nail on the head.

Thanks, I think it then comes down to each playing to its own strengths and not trying to be something else. Still, easier said than done for some reason.

 

[1oldman2]

There is no key or secret to understanding math. Teaching strategies that work for one teacher may not work for another because teaching is an interaction between people, and different people have different personalities, attitudes, expectations, etc.

American society needs to focus on empowering teachers and holding students accountable for learning. Currently neither of those things is happening to anywhere near a large enough extent. The problem gets worse as each generation of under-prepared students become the next generations' teachers.

Hopes placed in quick fixes like this are distractions from the much deeper issue that is the real problem.

Exactly what the "system" needs to hear, why isn't the system listening? I have a feeling we are battling a "Quantity over quality" approach here. along with a value system that may be lacking correct values.

 

[WWGD]

Exactly what the "system" needs to hear, why isn't the system listening? I have a feeling we are battling a "Quantity over quality" approach here. along with a value system that may be lacking correct values.

Sadly systems often tend to be more concerned with perpetuating themselves than with doing what they are intended to do.

 

[Mark44]

American society needs to focus on empowering teachers and holding students accountable for learning. Currently neither of those things is happening to anywhere near a large enough extent. The problem gets worse as each generation of under-prepared students become the next generations' teachers.

This problem seems borne out by the results of GRE exams for people with Ed degrees vs. all other degree fields.
Some figures that I have, from GRE scores for Education majors between 10/1981 and 9/1984:
Verbal ability: 444
Quant. ability: 470
Analytical ability: 403
These scores were the lowest of the 11 category groups students were divided into.These categories were: Humanities (included Arts, and Languages+Other Humanities); Social Sciences (included Education, Behavioral Science, Other Social Science); Biol. Sciences (inlcuded Bioscience, Health Science, Other Applied Bioscience); Physical Sciences (included Engineering, mathematics, and Physical Science)

The Verbal Ability scores ranged from 444 (Education) to 534 (Languages + Other Humanities). The Quant. Ability scores ranged from 470 (Education) to 667 (Engineering). The Analytical Abilities scores ranged from 403 (Education) to 574 (Physical Science).

These scores are dated, going back more than 30 years, but they are a snapshot in time, and I doubt that things have changed much in the intervening period.

Maybe the best you can aim for is to have teachers who are well-prepared and enthusiastic, who can transmit their enthusiasm to students through their teaching.

That is certainly an important part of the effort. Unfortunately the way administrators, parents, and students treat teachers, and the way students are not held accountable for learning interfere with that effort. Most teachers have their spirits broken. Or never consider adopting teaching as a profession in the first place because of these issues and the low pay.

In the private sector, employees are periodically evaluated, with the best employees promoted, and the worst employees let go. In contrast, in education, good teachers aren't rewarded, nor are poor teachers winnowed out. Teachers' unions (especially the NEA) vigorously fight any sort of evaluation of teachers. After an initial trial period, it's virtually impossible to fire an incompetent teacher. I speak from experience, having taught 2 years in a public high school, and 19 years at the college level. I also put in 16 years in the private sector, workiing at a large software firm in the Seattle area.

 

[WWGD]

This problem seems borne out by the results of GRE exams for people with Ed degrees vs. all other degree fields.
Some figures that I have, from GRE scores for Education majors between 10/1981 and 9/1984:
Verbal ability: 444
Quant. ability: 470
Analytical ability: 403
These scores were the lowest of the 11 category groups students were divided into.These categories were: Humanities (included Arts, and Languages+Other Humanities); Social Sciences (included Education, Behavioral Science, Other Social Science); Biol. Sciences (inlcuded Bioscience, Health Science, Other Applied Bioscience); Physical Sciences (included Engineering, mathematics, and Physical Science)

The Verbal Ability scores ranged from 444 (Education) to 534 (Languages + Other Humanities). The Quant. Ability scores ranged from 470 (Education) to 667 (Engineering). The Analytical Abilities scores ranged from 403 (Education) to 574 (Physical Science).

These scores are dated, going back more than 30 years, but they are a snapshot in time, and I doubt that things have changed much in the intervening period.

In the private sector, employees are periodically evaluated, with the best employees promoted, and the worst employees let go. In contrast, in education, good teachers aren't rewarded, nor are poor teachers winnowed out. Teachers' unions (especially the NEA) vigorously fight any sort of evaluation of teachers. After an initial trial period, it's virtually impossible to fire an incompetent teacher. I speak from experience, having taught 2 years in a public high school, and 19 years at the college level. I also put in 16 years in the private sector, workiing at a large software firm in the Seattle area.

I had , as an adjunct , a student-attendance rate of close to 100% (attendance not required in college) , good evaluations from teachers. No complaints filed against me by any student . But the a-hole chair did not like the way I dressed (though he himself looked like a lumberjack -- at the end of the day -- and even wore low-riders), so my contract was not renewed. As an adjunct I have very few rights, so I could do next to nothing and lost my position.

 

[Mark44]

I had , as an adjunct , a student-attendance rate of close to 100% (attendance not required in college) , good evaluations from teachers. No complaints filed against me by any student . But the a-hole chair did not like the way I dressed (though he himself looked like a lumberjack -- at the end of the day -- and even wore low-riders), so my contract was not renewed. As an adjunct I have very few rights, so I could do next to nothing and lost my position.

I'm sorry you lost your position. I was the head of the math department where I taught (community college) for 3 or 4 years, and it fell upon me to hire adjunct instructors, up to 25 of them in some quarters. There were too many of them for me to observe them individually, so I relied on student evaluations. There were a couple whose contracts I didn't renew, based on reports from students that I deemed were reliable and reasonable.

Your department's chair's complaint was about how you dressed. Can I ask what you wore that upset him so much?

 

[1oldman2]

What brought me to this thread was the subject title, As a student in 1-12 I stunk at math and always wondered why don't I get this. When asking an instructor for one on one help they seemed to always farm it out to another student who wasn't really concerned with whether I "got it" or not. (Don't get me wrong, mentoring is a great approach as long as the mentor and the student are engaged constructively). Later on in life I realized how really cool math is but the realization came from my own experiences on the job site and day to day life. Now I watch my granddaughter repeating my same experiences, I help her when I visit and she's very bright but the trick seems to be finding the right teaching approach on an individual level, not the mass production take that seems to be firmly entrenched nearly 50 years later. All of my grand kids are avid readers and that may be their saving grace in the education machine, at least they can visualize concepts rather than consume someones video versions without having to think for themselves.

 

[WWGD]

I'm sorry you lost your position. I was the head of the math department where I taught (community college) for 3 or 4 years, and it fell upon me to hire adjunct instructors, up to 25 of them in some quarters. There were too many of them for me to observe them individually, so I relied on student evaluations. There were a couple whose contracts I didn't renew, based on reports from students that I deemed were reliable and reasonable.

Your department's chair's complaint was about how you dressed. Can I ask what you wore that upset him so much?

No problem. I was kind of overweight (though not monstruously so) at the moment and the clothes did not fit me well. It seems like it is something that could have been easily changed. I also wore kind of ragged T-shirts, but always clean, without holes. Didn't know I was expected to be dressed up for a GQ photo shoot..

 

[WWGD]

What brought me to this thread was the subject title, As a student in 1-12 I stunk at math and always wondered why don't I get this. When asking an instructor for one on one help they seemed to always farm it out to another student who wasn't really concerned with whether I "got it" or not. (Don't get me wrong, mentoring is a great approach as long as the mentor and the student are engaged constructively). Later on in life I realized how really cool math is but the realization came from my own experiences on the job site and day to day life. Now I watch my granddaughter repeating my same experiences, I help her when I visit and she's very bright but the trick seems to be finding the right teaching approach on an individual level, not the mass production take that seems to be firmly entrenched nearly 50 years later. All of my grand kids are avid readers and that may be their saving grace in the education machine, at least they can visualize concepts rather than consume someones video versions without having to think for themselves.

Good point, learning what works for you -- in all sorts of areas: helth, emotional management, learning style -- may be the future of our education system. It would be great and fun if it happenned.

 

[1oldman2]

I had , as an adjunct , a student-attendance rate of close to 100% (attendance not required in college) , good evaluations from teachers. No complaints filed against me by any student . But the a-hole chair did not like the way I dressed (though he himself looked like a lumberjack -- at the end of the day -- and even wore low-riders), so my contract was not renewed. As an adjunct I have very few rights, so I could do next to nothing and lost my position.

I agree 100%, it seems once a teacher is vested "accountability" becomes a very ambiguous word.

 

[Mark44]

No problem. I was kind of overweight (though not monstruously so) at the moment and the clothes did not fit me well. It seems like it is something that could have been easily changed. I also wore kind of ragged T-shirts, but always clean, without holes. Didn't know I was expected to be dressed up for a GQ photo shoot..

It sounds like you might have run afoul of an implicit dress code, that didn't include ragged T-shirts. Where I was teaching, and when ('79 through '97) most of the male teachers wore regular pants (not jeans) and shirts (not T-shirts). There weren't any explicit dress-code rules spelled out anywhere, but most of us attempted to maintain a professional appearance.

 

[WWGD]

It sounds like you might have run afoul of an implicit dress code, that didn't include ragged T-shirts. Where I was teaching, and when ('79 through '97) most of the male teachers wore regular pants (not jeans) and shirts (not T-shirts). There weren't any explicit dress-code rules spelled out anywhere, but most of us attempted to maintain a professional appearance.

I thought Mathematicians were expected to be sort of slobs :) . Most others dressed similar to me, though. Most Math people I know dress way down, jeans and t-shirts. Professors wear dress pants and dress t-shirts, some wear sneakers. For some reason, applied Math people seem to dress better than those who do abstract Math.

 

[Mark44]

I thought Mathematicians were expected to be sort of slobs :)

Not those who are trying to keep an adjunct teaching position...

. Most others dressed similar to me, though. Most Math people I know dress way down, jeans and t-shirts. Professors wear dress pants and dress t-shirts, some wear sneakers. For some reason, applied Math people seem to dress better than those who do abstract Math.

 

[WWGD]

Not those who are trying to keep an adjunct teaching position...

10-4 , I am dressing more formally now. Although I suspect the clothes thing were a front for something else, but it gave him an excuse for firing me.

 

[Mark44]

10-4 , I am dressing more formally now. Although I suspect the clothes thing were a front for something else, but it gave him an excuse for firing me.

That certainly is a possibility.

 

[jerromyjon]

"Johnson's method relies on drilling"... that says it all for me. If you make games where you need the math you learned to win, you have motivation. The more you play the better you get... but the quality of what you are reinforcing is just as important. If I had the quality these kids deserve when I was in school, I am certain I would be better than I am at math! (not that I am NOT good, I was tortured for being the nerd with all the answers, I just never had a good teacher)

 

[gleem]

This month, she received a copyright for a diagnostic test that she says can assess specific gaps in students’ math knowledge in minutes.

She’s now working with the foundation to raise money to digitize the test, which includes eight to 10 math problems for each grade level, so that it can be used in schools throughout the country.

If this is true this could be extremely valuable and save much time and anguish in evaluating student's competency in math as well as helping to develop better curricula.

Johnson’s methods rely on drilling in the basic concepts of math — or, as she puts it, “the laws of math” — and ensuring that students understand why each step of solving a problem is necessary.

But isn't drilling frowned upon in our educational system?

 

[Mark44]

Johnson’s methods rely on drilling in the basic concepts of math — or, as she puts it, “the laws of math” — and ensuring that students understand why each step of solving a problem is necessary.

But isn't drilling frowned upon in our educational system?

Unfortunately, IMO, there are too many in the educational system who look disparagingly at drill. When I was teaching at a community college some years ago, a fellow math instructor never said "drill" without prefacing it as "vacuous drill." He said it so much that I abbreviated it for him as VD.

There was, and maybe still is, a movement to eliminate drill in the teaching of mathematics. This is extremely short-sighted in my view. Other endeavors, such as music and sports, require a lot of time to be spent on the fundamentals, essentially on drills to commit certain motions to what is called "muscle memory." If you have to think through each step of how to play a certain piano piece, or each step of a complicated football play, it will show that you haven't practiced these moves. The same is true in mathematics, I believe, going all the way back to being able to add or subtract single digit numbers and multiplying numbers at least up to 12 times 12. There were "educators" who said that students didn't need to know how to do these operations.

If you build a house without a good, solid foundation, the house won't last long. With vast numbers of US high school graduates who find themselves completely unprepared for college level courses in English, math, and the sciences, you have to question how solid their foundations in these subjects are.

 

[gleem]

After the launch of Sputnik an educational panic ensued leading to a seemingly endless futile 50+ year endeavor to improve our country's math prowess. During the period prior to the revolution in our math programs drilling was part and parcel of our educational system. Yet the products of this system despite our tardiness in launching a satellite were to produce the first commercial nuclear power plant, invent the transistor, the integrated circuit, the laser, put a man on the moon, string theory, and give us 43 Noble Prize winners in Physics. So what was the problem that we need to fix?

 

[bballwaterboy]

If this is true this could be extremely valuable and save much time and anguish in evaluating student's competency in math as well as helping to develop better curricula.
But isn't drilling frowned upon in our educational system?

I would love to see her "diagnostic" test too!

Although, I always have some built-in suspicions about things when people make "too good to be true" sorts of claims. Can a single test really be the solution to everything in math education?

As for drilling, I think drilling/practice helps with some things, but not others. You need to practice solving math problems, but you also have to understand the concepts behind them. The OP quote said she drilled concepts.

My calculus professor last semester had short-answer (expecting like three to five sentences) sections on some of our exams. He asked us to explain some concept or process in words, which required us to understand the logic behind what we were doing.

 

[davidhyte]

I agree that teaching the "laws of math" is crucial but we need to distinguish teaching the concept and teaching its name. You don't really need to talk about associativity or distributivity formally before you've actually shown the student or pupil that there are several cases where it holds and cases where it doesn't.

I believe a contemporary strategy of teaching should focus much more on presenting many examples where a particular pattern occurs, which I believe stimulates the student's creativity by allowing him to naturally imagine generalizations. Understanding the pattern and the reasoning applied to it is much more important nowadays than being able to quickly compute by hand some algorithm (like dividing numbers) because you can always (and you will) program a computer to do it for you.

For example, if you say that a number is a sequence of patterns that repeat themselves on a line (pick one of ten segments between 0 and 1 then pick another one within that segment and so on...) then you could also do the same for the 2D plane (pick a square within a square then pick another square within that square and so on...). You can come up with a dozen examples like that which stimulate the child's imagination. I've seen kids light up when they understand these concepts in a way they can use and have fun applying to things that have nothing to do with what you usually find in a math class. Kids love to cook up different concepts together to make new things.

Then once they are familiar with the pattern you could also tell them "oh and you know, people call that a positional system."