Is This the Key/Secret to Learning Math?

[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."

 

[Herman Trivilino]

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.

First the thing, then the name of the thing. Famously said repeatedly by Arnold Arons.

 

[Greg Bernhardt]

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.

My wife is a first year elementary school math teacher. Parents disrespect her, administration disrespects her, she works 10-12 hour days and makes a bit more than a fast food manager. My eyes have never been wider on the primary education system in my life. The tragedy is that is is amazing with kids and is a great teacher. Give her the support, the respect, the tools and she becomes a life changer for these kids.

 

[stevendaryl]

I guess because it's a pet topic of mine, what struck me strongest about the original post wasn't the "laws of mathematics" business, but this line:

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

I don't have any idea about how to teach math, but I do know what drives kids out of math completely, and that's exactly "gaps in student's math knowledge". My experience with trying to help kids with math, at least, shows me that so often, kids have trouble with math at one level because they have never completely understood the foundations that it is supposed to be built on. If you don't know how to add, subtract and multiply, then you're going to have trouble doing fractions. If you don't really feel comfortable with fractions, you're going to have trouble in trigonometry and algebra. If you really don't understand algebra, you're going to have an enormous trouble in learning calculus. Math in particular is cumulative, so a gap in fundamentals can haunt a student for the rest of his academic career.

​

 

[Herman Trivilino]

If you don't know how to add, subtract and multiply, then you're going to have trouble doing fractions. If you don't really feel comfortable with fractions, you're going to have trouble in trigonometry and algebra. If you really don't understand algebra, you're going to have an enormous trouble in learning calculus. Math in particular is cumulative, so a gap in fundamentals can haunt a student for the rest of his academic career.

The other side of that coin is, because of the cumulative nature, students have repeated opportunities to pick up those gaps that slipped through in the past. Many times students won't really learn a topic until they find they need it to learn another topic.

 

[ZapperZ]

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.

I completely disagree with this.

I hate math, and had always hated learning math. However, I was quite good at it in college, so much so that a few instructors thought that I should pursue a theoretical physics career. When I told them that I can't stand math, they were surprised.

So no, it is not a necessary criteria to "love it" to be good at it.

Zz.

 

[mrnike992]

Just to throw in my 2¢ worth here, I think there's a pretty obvious answer to why her methods are so successful.. I doubt it has much at all to do with her teaching style, and more to do with the fact that she has dedicated, one-on-one access with the under-performing students. If all teachers could work after school one-on-one, or even with smaller class sizes, I believe most would be capable of getting those students up to speed.

 

[Jeff Rosenbury]

I remember being taught in the electronics in the AIr Force (1980s). I asked our instructor how he did when he took the class. He said he had never taken it. He simply followed the T.O. (technical order) and his training in training others. In 16 weeks I picked up more technical knowledge than in two years of E.T. coursework. (There were other things learned in college, but for pure technical training, the Air Force taught what it needed to quickly and efficiently.)

The Air Force didn't think teaching was some big mystery. They didn't even require the teacher know the subject, just follow a prearranged lesson plan.

 

[chiro]

I sense that high schoolers interpret mathematics in a very different way to what should exist.

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation (again - in an organized and consistent way).

This is the real power of mathematics and I sense that the rules obfuscate this real understanding.

This is particularly notable when you look at normal mathematics problems. In the context they are presented the understanding of variation is obfuscated by ridiculous problems wasting both the teachers and students time and presented in such a disorganized and unconnected way that many students forget everything a couple of weeks into their final break.

Focusing on the rules per se doesn't get to understanding the variation as well as understanding how more importantly to think about how this variation can - and does apply, to the real world.

They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

I did a couple of weeks doing student teaching in a very good school and unfortunately I saw first hand just how bad this can be.

Instead of having mathematics being a used to understand variation and consistency in many ways - which is also a survival attribute when you realize that people are constantly bombarded with information, claims and logic in which they need to be able to sort the BS from the non-BS, mathematics is instead a bunch of disconnected and seemingly random (and pointless) ideas shoveled down kids throats for which many of them will soon forget and far more will never end up appreciating it (mathematics) for what its value is - including the ability to make sense of the world and be able to mount some sort of critical defense to all of the BS information that people have to navigate through and fight against.

This is what mathematics is about and this is where it's value lies - it lies in being able to look at variation and uncertainty and navigate through it in the best possible way - something which most high school students never end up figuring out - and partly because of how the subject has been stripped of its meaning and been used to facilitate lots of garbage that does the opposite to what it should do in terms of facilitating the above.

 

[UncertaintyAjay]

I completely disagree with this.

I hate math, and had always hated learning math. However, I was quite good at it in college, so much so that a few instructors thought that I should pursue a theoretical physics career. When I told them that I can't stand math, they were surprised.

So no, it is not a necessary criteria to "love it" to be good at it.

I certainly didn't mean that it is a necessary criteria. You don't need to love something to be good at it ( biology, in my case) but if you love something, you will try to be good at it.

 

[gleem]

Learning math or lack of is further exacerbated in the home. Parents are continually requested to participate in the education of their children. But math teaching techniques have become so unfamiliar when a child asks for help the parent and the child become frustrated. The parent not understanding or appreciating the technique may refuse to help leaving the child in a quandary, " how can I learn it if my parents cannot or will not help" or if the parent tries to help ends up either confusing the child or causing him/her to just shut down.

This can be improved by maintaining a consistent teaching technique over a span of time that includes the educational experience of the parent and the child, about 30 years. How many time have math programs changed in the last 30 years?

 

[clope023]

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".

Math follows the laws of logic last I checked.

 

[Jeff Rosenbury]

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".

Not only does math have laws, it also has regulations, goals, and prizes. Here is an http://www2.ed.gov/programs/racetothetop/executive-summary.pdf.

Math follows the laws of logic last I checked.

Not as far as I can tell. But I admit I've never understood bureaucrats, so maybe I'm mistaken.

 

[clope023]

Not as far as I can tell. But I admit I've never understood bureaucrats, so maybe I'm mistaken.

Cute, but I'm not referring to how math education is managed.

 

[Jeff Rosenbury]

Cute, but I'm not referring to how math education is managed.

Sorry; I've been struggling with concrete thinking.

 

[Mark44]

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation (again - in an organized and consistent way).

This description of what mathematics does and how it should be used is so high-level (a "50,000 foot view"), that is not very useful, IMO.

This is particularly notable when you look at normal mathematics problems. In the context they are presented the understanding of variation is obfuscated by ridiculous problems wasting both the teachers and students time and presented in such a disorganized and unconnected way that many students forget everything a couple of weeks into their final break.

I'm not convinced that an understanding of variation is important. Maybe you can give some examples of what you mean. I agree that concepts need to be organized, with connected themes running through the concepts, and that problems that waste time should be eliminated, but could you elaborate on the kinds of problems you're talking about?

Focusing on the rules per se doesn't get to understanding the variation as well as understanding how more importantly to think about how this variation can - and does apply, to the real world.

They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

Are you arguing against the memorization of these concepts? If so, I strongly disagree, as these are the fundamental concepts that need to be in a student's "toolbox" so that he/she can tackle applied problems that use these concepts.

Going back to my earlier analogies of music and sports, if a guitar player hasn't spent many hours learning how to shape (for example) a Bm chord followed quickly by D and A chords, the song being played won't sound good. And similarly, if each player in a football offensive team hasn't spent many hours committing each play to memory, the outcome for that team is not favorable. Why would things be different in the teaching of mathematics or any other academic study?

If a student in physics doesn't have the sine, cosine, and tangent functions and quadratic formula committed to memory, said student will not likely be able to even start applied problems involving multiple forces acting on an object, or involving an object that is thrown through the air.

You mentioned "understanding the variation" several times, so I gather that it is important to you. You didn't expand on what this means to you, but by itself, I don't see how this understanding is helpful to students of mathematics.

 

[Student100]

Math follows the laws of logic last I checked.

There's no such thing. This is a philosophy question, so I won't get into it. The word law shouldn't exist in the formal/natural sciences. Again, if you saw my earlier post, we could argue semantics all day long but I conceded it isn't useful to the thread.

Not only does math have laws, it also has regulations, goals, and prizes. Here is an http://www2.ed.gov/programs/racetothetop/executive-summary.pdf.

There are lots of those laws, I must agree.

 

[Mark44]

Math follows the laws of logic last I checked.

There's no such thing. This is a philosophy question, so I won't get into it.

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

The word law shouldn't exist in the formal/natural sciences.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

 

[clope023]

There's no such thing. This is a philosophy question, so I won't get into it. The word law shouldn't exist in the formal/natural sciences. Again, if you saw my earlier post, we could argue semantics all day long but I conceded it isn't useful to the thread.

The word law might be a mis-nomer but they tend to be followed as such; 'Law's' of non-contradiction, causality, and such like are absolutely followed in math and physics.

 

[Student100]

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

I am objecting to the word law, of course the foundations of mathematics follow logic.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

It does more harm than good. I understand, and you understand, what the context of the word is. Many students, especially in high school and introductory science courses, don't. It leads to confusion about what science is actually trying to do, what models actually say, and belief that scientific "laws" are somehow infallible simply due to a poor choice of words.

The word law might be a mis-nomer

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

 

[Herman Trivilino]

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation [...]
They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

I agree completely. This situation is a result of the way our American society has attempted to remedy the very problem we've created. We have failed to support and respect our teachers, and we have failed to hold our children accountable for learning. When confronted with the evidence of poor student performance in comparison to other countries, our response has been what I call educationism. We attempt to hold our teachers accountable by dissecting their subjects into pieces, followed by measurements of how well students perform each piece. So we end up with a lot of teachers who teach only the pieces. This seems to be especially true of mathematics.

The latest version of educationism is placing an emphasis on student learning outcomes (SLO's). By listing the SLO's associated with the most populated college courses taught in the state of Texas, the Texas Higher Education Coordinating Board (THECB) has now assured us that each course is equivalent, regardless of the instructor or college where it's taught. Administrators at these colleges, in response, are now making instructors not only list these SLO's in their course syllabi, but show evidence, called assessment data, that each SLO is being addressed in the teaching of the course. And that each instructor is assessing each student's performance on each SLO. This will further assure us that these courses are equivalent. Of course, if the data show poor student performance, the instructor is supposed to make improvements to the curriculum, the teaching methods, or the assessment methods. If the data show good student performance the instructor is supposed to make improvements to the assessment methods because, presumably, the assessment instrument is not rigorous enough.

With my assessment data I have been able to show that everything is good and nothing is bad. As this process was becoming institutionalized during the last decade I warned anyone within earshot that the system would never measure anything of value. I stated obvious things that any instructor could do to produce good assessment data in the absence of good teaching and good learning. I was criticized as someone who is really good at coming up with ways to beat a system. So, I stopped.

 

[Rx7man]

I'm extremely surprised no one has brought this KEY failure in the educational system

Boredom!

This is especially true for brighter students.. They 'get' the concept the first time around... but they spend the entire rest of the year rehashing it.. by the end of the year, they're bored out of their minds, they've only learned half of what the could have learned, and have lost motivation.
For me, this was particularly true of English class... I think it was from about grade 2 to grade 10, I pretty much hashed and rehashed ad infinitum what verbs, nouns, and adjectives are.. 8 freaking years of this.. I always did poorly in English, it was an absolute bore.. Then I finally took English 101 in college, most of my classmates were Asian imports, and the teacher didn't really pay a whole lot of attention to sentence structure, etc.. he focused on reading comprehension, etc.. I actually got a B in that class!
I was homeschooled for grades 4-8, then I went to an unaccredited academy for 9-11... They had a terrible teaching program, and I ended up redoing grade 11 in public school. I was placed in Math 11, with about Math 8 skills, I had NO concept of algebra and my fractions were pretty shakey. I had a great teacher and she spend a little time with me to get my feet back under me.. We were on the quarter system, so each quarter was about 10 weeks of school with 2 classes per quarter.. It meant I studied each of the 2 subjects for 3 hours a day.. this was good, I was able to immerse myself into the subject and get to the bottom of it.. I think I had a high C.. I took Math 12 the next quarter and got an A, In the other science courses I took I got 98% in Physics 11, 90% in Calculus 12 (an elective), and 94% in Biology 12 (beating the teachers daughter).. I was interested and excited about the subjects and I applied myself
Then came university.. Like most people, I got distracted, class sizes were about 300 students, I was shy, so didn't really know anyone, which also meant I didn't have any 'competitors'... I was sedentary, I couldn't do any of the things I loved doing (fiddling with mechanical stuff, cattle, dirt bike rides,etc) and so I ended up just getting a string of dead-end jobs to pay bills.I think the educational system has to get it's priorities straight.. Give the kids who smoke weed behind the school enough skills to count out the dime bag, and leave it at that.. stop trying to make physicists out of them... The kids who want to learn, fast track them whenever you can.. don't bore them into smoking weed behind the school!

 

[Mark44]

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

I am objecting to the word law, of course the foundations of mathematics follow logic.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

It does more harm than good. I understand, and you understand, what the context of the word is. Many students, especially in high school and introductory science courses, don't. It leads to confusion about what science is actually trying to do, what models actually say, and belief that scientific "laws" are somehow infallible simply due to a poor choice of words.

I don't believe that changing the word "law" to "property" or "principle" (or whatever) would decrease the confusion. The problem is, I believe, in not listing the conditions under which the law can be applied. Newton's Second Law of Motion is F = ma, which is not valid for an object whose mass is changing.

In any case, there are many laws that are named after the persons who discovered them. See the wiki article for a long list of such laws: https://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people. It makes no sense to me to revise history by renaming, say, Kirchhoff's Law or Boyle's Law.

The word law might be a mis-nomer

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

It seems to me that your disagreement is due to the way these "laws" are presented (or misrepresented) rather than with the use of the word "law" per se; i.e., without any "fine print" giving the limitations. One example is saying that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ without also listing the restrictions on a and b.

Also, do you have any evidence that she is inaccurately conveying these laws? Just because some teachers and some references are sloppy doesn't mean you can extrapolate this sloppiness to every teacher.

 

[MidgetDwarf]

I think the secret is having knowledgeable teachers in the classroom and changing the culture between: parents, students, and administration. Of major in importance is access to textbooks for students. Many children still do not have textbooks for classroom use and take home purposes. I know atleast 20 people who struggled with math, statistics were there highest math class taken, majored in the arts, and who are now teaching mathematics to young students.

 

[Herman Trivilino]

I'm extremely surprised no one has brought this KEY failure in the educational system

Boredom!

That's not a failure of the educational system. It's a failure of our society. Students who are bored with learning could be removed from those classes in which they're bored and put to work doing something productive. But our society will not allow that. Instead we as parents and educators coddle these "bored" students and pass them along to the next grade, where of course they continue to be bored.

If you're bored, guess what? It's not anyone else's fault. No one else is responsible for your failures or your emotional state. Find something interesting to do, a way to do it, and stop whining.

 

[clope023]

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

People can't read and look up the nuance when they get more mathematically mature?