If basic math skills of a student are weak, has he/she an opportunity of mastering more complicated topics?

Here are the total results of the poll on LinkedIn (six groups):

“If basic math skills of a student are weak, has he/she an opportunity of mastering more complicated topics?”

Yes   55 (44.4%)         No   69 (55.6%)

I am surprised at these results. Compare them with the results of the previous poll "Are you satisfied with basic math skills of your students?"

Yes   9 (8.7%)           No   94 (91.3%)

My experience shows that weak basic skills bring about failure in further math learning (http://www.simplar.narod.ru/Victor_Guskov.doc). Maybe just the opinion that there is an opportunity of successful mastering more complicated topics without solid fundamentals brings about so sorry state of students' basic math skills?

Further on you can read some interesting comments which have been separated into three groups

1) YES

2) BUTS AND IFS

3) NO.

YES

Carol Nicholson •  A student that has trouble mastering multiplying or dividing decimals can easily understand concepts such as slope of a line, linear equations, etc.

Joy McCutcheon • I have had many students to grasp the "algebra" that make simple calculation errors due to being weak on the basics.

Matt Tucker • If we are saying basic skills= number sense....of course students can take on more complicated tasks. My top honors geometry student had horrible basic computation skills, but would persevere through difficult proofs. We all have strengths and weaknesses.

Allen Macfarlane • I voted a yes, based on my high school math experience. I was a C student in math and did not do that well in algebra. However, as a senior I took a course in trigonometry in my first semester and earned an A. I loved doing the proofs and also applying trigonometry to practical problems, something that was lacking in my algebra and geometry classes.

Julia Brodsky • Times and times again I teach profound math concepts to young children, who are still learning their multiplication skills - and they are enjoying it. Deep topics such as infinity, game theory, graph theory, symmetry, logic and numerous others can be presented to children with very limited arithmetic skills. Calculus can be taught to very young kids, as demonstrated by Don Cohen and others. Math circles are a living example that math is not linear, and can be approached in many different ways.

Cecilia Villabona • What do we mean by "basic skills" in mathematics? How do we present the concepts to the learners can greatly affect his/her understanding.

It is possible to teach the basic ideas and concepts of calculus to very young children, they still need the algebraic fluency to really do derivatives and integrals. In this example, the "basic math" is Algebra.

I worked with teachers who often used the phrase: "My students don't know their basic Math, so I can't teach them anything" and they were constantly frustrated trying to teach junior high or high school students how to do long division, or their preferred method to reduce fractions to a common denominator. This produces a lot of frustration for both teachers and students, and yes it becomes impossible to teach anything else because know you have no time, and lots of discipline problems.

So to go back to Victor's question: with good materials and good pedagogy you can teach children the math topics, regardless of their prior knowledge. If students have number sense, they can use calculators effectively for concept development, and with computer aided instruction, students can work individually at their missing "skills" to help them with fluency.

Joseph Austin • As I recall, I "mastered" the material in Math course N when taking Math course N+1:

I mastered Algebra in Trig, Trig in Calculus, calculus in Physics, etc.

I think the opportunity to repeatedly use the material in applications is a good way to master what may have been sketchy on first exposure.

My other concern would be, what do you consider "basic"? The more "advanced" my classes got, the more rigorously they covered the "basics".

I'm assuming we are discussing a student who wants to advance in spite of a checkered past. I would let the student try the advanced level, then point them to resources to reinforce weak areas. Motivation can trump talent.

Ben Woodford • I am with Joseph Austin. Complicated ideas by default reinforce the basics. Many students, myself included, do not take well to memorization and tedious repetition. When they use the previously fuzzy concepts to solve a more complex problem, students can learn the importance of the basics they didn't grasp previously.

I believe a student can solve equations without knowing negatives. In fact, the act of trying to solve an equation would necessitate the use of a subtraction or division by negative and therefore give the student a chance to discover the use (fullness) of negatives. As a facilitator we can suggest the use of the (-), as a symbol to perform the students required task.

Isn't discovering a problem and inventing a method to achieve that task, the job/pride/joy of mathematics since the time of Pythagoras?

BUTS AND IFS

Mister Jones • The question's wording allows for many more yeses, but I get the jist, anyway. It fully depends on what "basic" Math concepts we're talking about. Many of the "basic" Mathematical concepts are prerequisites for fully understanding and applying (read: mastering) more complicated concepts. Multiplication is a basic concept, but Vertex Edge Graphs are more complicated. However, a student needs not be able to have strong computation skills in order to accurately read or "solve" a vertex Edge Map. What it boils down to is that Mathematically, like many other subjects, the bigger the base of the building, the higher the building can be built. That's my story and I'm stickin' to it.

Bonnie Yelverton • I agree with Mr. Jones that the wording forces a Yes. It should have been something like "have difficulty mastering more complicated topics." Some topics include more basic (by which I understand PreAlgebra) skills than others. But there are ratios everywhere, and if a student doesn't understand fractions, he's going to have trouble with similarity, etc.

But opportunity, yes, because it doesn't mean he can't learn those skills. It's just that we teachers have to help him get them up to speed so he can go on, as soon as we figure it that is what's holding him back. (And the earlier teachers discover that, the better!) As someone said, students may have life experiences that distract right when they're supposed to be learning something important and basic.

Irene Sawchyn • The question is a loaded one - WHY are the basic skills weak? If the question was worded, "If a student cannot develop basic skills...” then my answer would be NO. If the student has had poor training, then my answer would be YES, after they received better training.

I see that middle- and high-school students who come from eastern Europe or Asia come 1-2 years ahead of their American peers in math. How could that be? They just never hear the words "is the student "ready" for math (at some level)". I am the eternal optimist and believe that (almost) all "average" students should be able to master HS math if prepared well in grade school, and dedicate enough time to the work.

James Smith • I have to agree with Irene. This a "When did you stop beating your wife?" type of question. If the job of educators is to educate, the belief exists that this weakness will be seen and then addressed, hopefully in a way that will increase the skills. I have had this on my own experience and it means you must be willing to use your curriculum to provide support. Not an easy task, but one an educator must take on, if they are to ensure future success in their own subject.

I am not sure that one can quickly teach, and have students master, the basic maths necessary to be successful in chemistry, or physics, but, if I have any integrity as an educator, I must be willing to stop my own curriculum progression and deal with the defect. If I ignore it, the class can be irreparably damaged.

Laurence Cuffe • In most cases, don't run before you can walk. I can however imagine a case where I might teach some functional maths as part of life skills basing the computational aspects on the use of a calculator to someone whose dyscalculia rendered it unlikely that they would ever master paper based computational algorithms.

James Rasure • Unless you consider going back and fixing the poor math skills as an opportunity. For example: I teach basic math at a community college. Some of the students have horrible math skills which would keep them out of a nursing program. They take my class which gives them some basic math skills and more. Thus they have the OPPORTUNITY to catch up and master more complicated topics.

Diane Sanders • This question has interesting phrasing. "Opportunity" - always. They have to do what James suggests - strengthen the basics.

Math skills being "weak" - I have found poor grades can sometimes reflect mere laziness - a preponderance of careless errors.

"complicated topics" - Visual or mechanical/procedural learners are sometimes surprising - someone who can't calculate correctly trig functions can memorize the shape of the curves and answer qualitative questions well, and vice versa.

But the key word you use is "master". I wouldn't say merely remembering procedures or formulas or shapes of curves is mastery, even if it leads to a high score when tested. No, complicated topics won't be mastered without strong fundamentals. But barring a learning deficiency, one always has the opportunity to master the fundamentals and build from there.

Doug Hainline • We need to analyze the original question and unpack its intended meanings.

(1) By 'a student' is meant, presumably, 'any student', or 'all students'. That is, no exceptions.

(2) By 'cannot', is implied, the student has tried very hard, has had a good teacher, but due to some internal reason - e.g. low IQ -- is unable to master basic skills.

Several people here have argued that they personally know of exceptions. Assuming that these exceptions fit the second 'unpacking' above (2) -- that is, they are actually unable to master certain basic mathematical skills, but can master others -- then that would seem to close the discussion, assuming that (1) was meant.

If however, the question was meant to be something like a weaker variant, e.g. "Is it generally the case that a student who has not (for whatever reason) mastered basic skills, will not master advanced ones", then probably most of us would assent.

Dave Hinz • What comprises basic math skills? I had students who did not know their multiplication tables in high school. They could get around that shortcoming by using a calculator, although they still had to understand hierarchy of operations. These same students had difficulty comparing fractions because they could not find the LCM. They could convert the fractions to decimals and compare them, again, using the calculator. I think these students had more difficulty with all math topics beyond basic such as exponents, scientific notation, lines, slopes and inverses. Factoring polynomials is easier if you know the multiplication tables. That may be the key. They can probably "master" more complex areas of math but it will be harder and it will take them much longer. That may be enough for them to give up trying.

What is mastering? You can load data into a program and perform statistical analysis with little knowledge of math. Do you understand the output? You can find roots of polynomials and solve series of linear equations with matrices using graphing calculators. This is not mastering, but it may be adequate for many business requirements.

NO

Ananya Mishra • No, The basics and fundamental must be strong before attempting complicated topics....

1. • Not possible. He can go only for simple applications but higher grade math requires basic concepts well. Advanced math is not for all.

A V Prakasam • A strong foundation and basic knowledge are very important for erecting further building blocks. This is true not only for math but also for other science subjects.

Joyline Maenzanise • The basic skills lay the foundation for the complicated concepts. So, if one struggles to grasp the simple concepts, then it goes without saying that he/she may have difficulties in understanding the hard stuff.

Charles Carter Waid • Learning your math facts is akin to learning your scales in music. You can't play math without it. I never met a child who is confident in math that didn't know their math facts.

Paul Ferrino • This question is loaded but overall I would answer, "No." Students who have not mastered basic concepts will struggle in more advanced topics. I base my answer on experience teaching kids high school and middle school kids in an urban school district.

Richard Jensen • When you consider that the basic math skills are adding, subtracting, multiplying, and dividing with integers and rational numbers, there is little chance that someone weak in any of these areas will be successful in the more complicated topics.

Carlton Johnson • It goes without saying that Mathematics - like other disciplines - builds upon earlier concepts. Students are first taught addition, then multiplication, then exponential functions etc. That said there are concepts in Geometry - for example - which require more visual/spatial recognition than numerical manipulation required for Geometry. However, beyond that one subject if a student were to go any further i.e. in college, the student - I believe - would find it difficult to succeed unless he/she had first succeeded in the fundamentals.

ramanpreet bhatia •  Its a very good question as many of the students feel that they should be taught only concepts related to the curriculum of that grade when I used to have a recap of basic skills. I think it’s our responsibility as a teacher to reinforce the basic skills in our students. A strong building is always on a strong foundation. I feel if they have good basic skills then only they will be able to see and enjoy beauty of complexity in mathematics.

Richard Catterall •  In my limited experience teaching mathematics, only 32 years so far, I have found that students who know their multiplication tables instantaneously (as opposed to sometimes, on a good day) have a higher probability of learning higher mathematics than those who do not. I encourage parents to encourage their children to learn the times tables, as late in their learning careers as many of them are, and by bribery if they cannot think of a better method (money seems to work quite well). We all know that you can lead a horse to water ... and our job as teachers is to give them salt!

George Barboza • Even in Mr. Catterall's "limited experience" I see a sense of wisdom that I certainly couldn't agree more with. Even for those who think that requiring "memory of trivia" such as multiplication tables is not a good way to teach mathematics, simply since the analytical factor is more important, should remember that without memory of basics you not only don't have much success in mathematics complex concepts, you don't have much success in life in general. If memory of basic things is not important, try not remembering your name. See how far that simple lack of basic memory takes you. The same analogue applies to mathematics, even more so, because of the building block basis of ALL mathematics. Enlightening piece Richard.

Dan Umbarger • My take is that when the high school student has not mastered their multiplication tables and are watching a teacher work a problem (presumably without a calculator) then they really are unable to follow his/her logic. Over time this lack of understanding what is going on accumulates until math becomes a mystery rather than a beautifully sequenced logic.

The math guru at my previous district always signed her emails "Life is too short to spend on long division!" Under her leadership the students failure to learn traditional elementary curriculum were covered up by "social promotion" pass/fail quotas and when the students graduated and took placement tests at their post high school institutions they found out that they were functioning at 4th-8th grade levels so that is where their community colleges placed them. But hey! They had a high school diploma.

Anthony Emmons • The key word here is master. Without strong basic skills, a student can grasp/understand the concept. But being able to fully apply the concept is another issue. For example, the concept of a derivative is actually quite simple. In-depth conversations can take place just by looking at graphs. You can define intervals of increasing and decreasing and other characteristics, again, just by looking at graphs. Moving away from graphs and trying to do the same with function rules is not so easy. Without the prerequisite skills of algebra, of line-by-line arithmetic, equation and inequality solving, factoring, etc, mastery of the concept of a derivative will not be achieved.

Raymond Griffith • If a student doesn't reinforce weak areas, can he/she succeed at advanced level?

The answer is no, not really. Some "dumbing-down" of courses have been done in order to make it appear that success was being achieved, but it really wasn't.

A student who is weak in arithmetic will have a horrible time in algebra. Algebra requires arithmetic. The better a student can add, subtract, multiply, divide and operate with fractions, the easier they will work with variables, functions, and the arithmetic operations we do with those. Many students miss problems they would otherwise get because they do not do the arithmetic correctly.

As an assist, we now have calculators everywhere! And for the most part, I don't mind. But the fact is that the best work still comes from people who can *do* the arithmetic, not relying upon a calculator. It turns out that you need some skill in subtracting to be able to know when to subtract. You need skill in multiplying to know when to do it.

As an analogy, think about a car. You need your brakes fixed. So you have shop number 1 where the mechanics have used their tools in all sorts of situations and tight spots. But they have not memorized the exact procedures for this style of brake pads. In shop 2, the mechanics can all tell you the exact procedures for replacing your brake pads, but they haven't practiced using their tools nearly as much as they have spent time memorizing the processes.

Which shop would you use?

It is the same thing in mathematics. Learning at a lower level enables learning at a higher level. Learning at a lower level well often allows ingenuity in solving higher level problems.