How to Fix Our Math Education - NYTimes

[ryoder]

I agree with you about needing both application and theory.

I think we should double down on math and science. Lets take the typical 45 minute algebra course and make it 90 minutes. That way students drill the basics, learn the theory, learn the history and learn how to apply it to real world scenarios.
I think that math and science should be at the base of the arts and sciences food pyramid after a reasonable reading level has been achieved.
My ideal situation is to get kids reading on a 5th grade level by the end of first grade at all costs. Once they have reading down, then concentrate on math and science. Reading is so key to becoming a good learner so lets focus on that first and then taper off and focus on math and science later on.

I also agree that people want to graduate with a CIS or MIS degree without actually learning how to write a computer program. This is ludicrous. These people often end up in business and systems analysis positions making decisions about what software to buy, what software to build, and how to build that software. It is unrealistic to think that the head surgeon at a hospital has never performed or even taken a class on surgery but you see this all the time in the information systems world. People with no understanding of how systems actually work end up dictating their design. This is very common in the federal and state government levels also, where career government employees direct the efforts of contractors with little knowledge of how to build computer systems.

[OE800_85]

Respect to the food pyramid analogy, I like that one. As well, THANK YOU to the earlier poster who made the post I kinda wanted to do, but was way too lazy to put together.

Also, someone earlier was wondering why other countries are outperforming America in STEM. You mean India and China? It's because if they don't get it right, they don't eat and their parents beat them until they do, lol. Despite the laugh that's not too far from the truth.

Wanna know my opinion? We pamper our kids. We baby them into oblivion. I remember one interesting survey from the movie Waiting for Superman, where American kids ranked highest academically in one category, their own personal evaluation of how well they did. They were well behind in basically everything else.
Without trying to be racist, there's a reason why Asians do better in school (particularly in STEM, where drilling is very effective). It's not biological, it's cultural. Parents, teachers, authority figures are tougher and expect more from kids. We should learn from this.

[OE800_85]

More of what I wanted to say, too lazy yet again:

"Context and applications are fine, but should be used to motivate abstract mathematical and statistical principles. The same exponential function arising in finance in compound interest calculations is needed in the life sciences to explain population growth. There is intrinsic and practical value to struggling to understand the mathematician's elusive "x", and to understanding the need to work with nondimensionalized equations when expressing general theoretical relationships between real-world quantities. The proposed pedagogical approach sounds a bit too much like dumbing down the curriculum."
Rob
Amherst, MA
August 25th, 2011
7:45 am

[OE800_85]

sorry to hammer this thread, but you guys really should all read the comments for this article, much more enlightening than the article itself, in my opinion

[Kaz]

The thing that really blows my mind is how no one seems to agree on the best way to go about it. I mean career teachers, PhDs, etc. You find all sorts on every side. Maybe there is no "best" way? This TED talk was pretty good:
Dan Meyer: Math class needs a makeover | Video on TED.com

[irnbru]

The thing that really blows my mind is how no one seems to agree on the best way to go about it. I mean career teachers, PhDs, etc. You find all sorts on every side. Maybe there is no "best" way? This TED talk was pretty good:
Dan Meyer: Math class needs a makeover | Video on TED.com

The issue is that it is deeply political and depends on what goals are to be achieved.

On the one hand, we could make maths classes more 'relevant' by making them easier so that everyone feels they are getting something from it by concentrating on a few applied models and examples. This would concurrently hold back the more able pupils. Parents wouldn't have to help their kids study, a system would be in place and education could be streamlined to produce 'useful' qualifications as preparation for the 'real world'.

On the other hand, we could make maths classes more traditional. A solid emphasis on arithmetic, geometry, etc AND 'modern' maths, i.e. a bit of discrete maths/group theory. This will lead to more students failing and parents having to help their kids.

The third option which falls out of a sort of synthesis of the first two is that we have a tiered system. While this is meritocratic, it creates problems with parents wanting their kids to be in x class in the Forest Gump 'my kid will have all the opportunities' sense.

Maths is really well suited to standardised testing but there will always be arguments along the lines of 'not a good test taker', 'not fair for one day' and so on. SATs and GREs pick up kids between highschool and college but there's not so much before, yet.

I think it's fair to look at standardised testing as equivalent to some forms of professional certification. If you can pass the test at a particular level, then you're in. If you can't pass the test, you can resit later but the sooner it gets passed the better.

I probably wouldn't pass a lot of these tests at a high level but I'm comfortable with that. It really depends how parents want their kids to feel about themselves and how much parents can help their kids (time, money, own education). The social ethics of emergent class systems already acknowledges the shape of opportunity through the generations, but I'd rather a reasonably meritocratic system than the nation dropping its standards.

edit: There's a really contentious debate extant at the cultural level here. One of the things I like best about the US is an attitude which isn't afraid of winning. This has to be balanced with accepting that not everyone is going to be a brilliant student. How to go about this in education is a tough discussion when opportunity is (and should be) a driving value. The same debate occurs in the UK with the abolition of grammar schools (and the recurrent moves to bring them back) and grade inflation. There's the problem of how students are streamed, why they are streamed, how universities can select the best students (especially when there are more 'perfect scores' than available places) and what the purpose of school should be.

[dcan]

My ideal situation is to get kids reading on a 5th grade level by the end of first grade at all costs. Once they have reading down, then concentrate on math and science. Reading is so key to becoming a good learner so lets focus on that first and then taper off and focus on math and science later on.

:iagree: I've been saying virtually the same thing for years now. Grade school should be focused heavily on reading, especially critical thinking and analysis, and writing. Too many kids today can't form a coherent thought beyond SMS/twitter-ese.

Regarding changing math curricula and "pure" math being "better" than applied math, this writeup from I think the 2004/5-ish timeframe may be informative:

The Mathematical Association of America
Renewal of College Algebra

The need for mathematical and statistical understanding by students in disciplines traditionally considered non-quantitative has grown enormously, yet far too many students are unsuccessful in traditional college algebra courses, or are unable to apply skills learned in the course in the context of other fields. There is some preliminary evidence that modeling-oriented, application-based college algebra courses that take advantage of modern computational tools may better-serve students than the traditional college algebra course.

...

The college and university community of mathematics faculty know that college algebra needs to be changed. The importance of college algebra is clear, if for no other reason that it enrolls over 1,000,000 college students each year. [1] The need for mathematical and statistical understanding by students in disciplines traditionally considered non-quantitative has grown enormously, partially as a consequence of the integration of computers and other technology into almost every facet of our lives. These technological devices bring powerful computational tools to our desktops, but appropriate and effective application of these tools requires basic algebraic facility and a deep conceptual understanding.

Yet far too many students are unsuccessful in traditional college algebra courses (with “DFW” rates at many institutions at or above 50%), or find that, in subsequent courses where quantitative skills are called for, they are unable to apply skills learned in the course in the context of other fields. Many leaders of collegiate mathematics have discussed the potential for modeling/discovery-oriented, application-based college algebra courses that take advantage of modern computational tools to be much more valuable for students than a traditional, skill-based, pencil-and-paper college algebra course. Through the MAA’s Committee on Curriculum Renewal Across the First Two Years (CRAFTY) Curriculum Foundation Project, weekend “conversations” among mathematicians and groups of faculty from 18 different disciplines were convened to determine what faculty in the partner disciplines hoped students would learn in their lower division mathematics courses. As summarized in the 2004 MAA report, Voices of the Partner Disciplines,[2] the faculty in the other disciplines recommended that mathematics departments should “Replace traditional college algebra courses with courses stressing problem solving, mathematical modeling, descriptive statistics, and applications in the appropriate technical areas” and “de-emphasize intricate algebraic manipulations.”

...

Approaches to be Adopted/Adapted/Implemented

The mathematics departments of the eleven participating institutions have agreed to pilot sections of a college algebra course with the following features: the central theme/organizing idea of the course will be mathematical modeling; students will be assigned long-term project(s); students will be assigned work to be completed in collaboration with other students; graphing calculators and/or computer utilities will be utilized throughout the course; the algebraic skills deemed by the institution as critical will be maintained in the course, but will be de-emphasized. The goal is to enable students to use mathematics to quantify real-world situations – that is, to mathematically model physical phenomena.

Source: http://math.arizona.edu/~laetsch/CRAFTY/...m-narr.pdf

I have such a textbook on the way from Amazon right now. I'm very interested in seeing how it takes what I've learned through ALEKS and applies it to real-world problems. You can "look inside" and use the "surprise me" feature to skip around and see how it teaches different concepts.

Here's another book I intend to get after doing trig: Functions Modeling Change: A Preparation for Calculus. From the description:

This is a new edition of the precalculus text developed by the Consortium based at Harvard University and funded by a National Science Foundation Grant. The text is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds. It provides numerical and graphical approaches as well as algebraic approaches to give students another way of mastering the material. This approach encourages students to persist, thereby lowering failure rates. A large number of real-world examples and problems enable students to create mathematical models that will help them understand the world in which they live.

The focus is on those topics that are essential to the study of calculus and these topics are treated in depth.

• Linear, exponential, power, and periodic functions are introduced before polynomial and rational functions to take advantage of their use to model physical phenomena.
  
• Building on the Consortium's Rule of Four: Each function is represented symbolically, numerically, graphically, and verbally where appropriate.
  

I read a blog post from the Math Association that said College Algebra needed to change from it's 1950s focus as a stepping stone to Calculus (which less than 10% of students ever even attempt) to something more relevant for today's business and technology professionals to apply to their daily work and personal lives. Not a "dumbing down" (the same concepts are still taught) but a shift in focus.

[irnbru]

Regarding changing math curricula and "pure" math being "better" than applied math, this writeup from I think the 2004/5-ish timeframe may be informative:

Hi, dcan,

The articles says two things:

1. Soft science and non-calculus subject faculties are talking about replacing pure maths courses with applied courses.
2. The content should be easier so more time is spent practicing simple techniques that these other classes use.

Key citation from your quote I think is:

Voices of the Partner Disciplines,[2] the faculty in the other disciplines recommended that mathematics departments should “Replace traditional college algebra courses with courses stressing problem solving, mathematical modeling, descriptive statistics, and applications in the appropriate technical areas” and “de-emphasize intricate algebraic manipulations.”

These are not so much your physics, chemistry depts but more your social science, education, business depts, etc saying that maths classes should be less about maths and more about practicing their problems, i.e. more time spent doing easier maths = dumbing down.

edit: Again, it all depends on what the goals are. The crux is that the core College Algebra course should be scrapped for general students and replaced with a course called something like 'Math Applications'. Bear in mind this is a college course and does not apply to high school students. Further, that any students in the hard sciences (i.e. physics) or engineering will still need to go on and study Calculus I, II, III, Algebra, Statistics, etc anyway. It is basically stating that College Algebra should be replaced with a remedial maths course. So, we can assess that some colleges are not receiving maths-literate students from highschool, which means that the highschool curriculum is insufficient preparation for college.

This is why I would prefer to see the work done earlier, rather than a slippery shift in tertiary education trickling down into high school to middle school and an overall lowering of standards.

[ryoder]

The social science departments are always looking to eclipse math and hard science. You have to watch out for them.
They even go so far as to say that math as we know it is racist! Some of the hard left have succeeded in replacing real math classes with socially responsible, watered down classes to the detriment of student learning.

Gene Expression: Anti-Racist Multicultural Math
Anti-racist mathematics - Wikipedia, the free encyclopedia

Critical thinking is also on the chopping block because it is seen as being sexist.
GENDER BIAS IN CRITICAL THINKING: CONTINUING THE DIALOGUE - Wheary - 2005 - Educational Theory - Wiley Online Library
I ran into this crap when I was writing a paper about critical thinking for my MBA class. There are those who believe that the purely reasoned rational thought process that is required for critical thinking is biased toward men who are rational and guided by universal principles.
Women, they argue, are less rational, more emotional, caring and write in a personal tone.

If you read a couple hundred scholarly journal articles like I have had to do in the past few months, you will see critical thinking at work. There is no place in scholarly writing for unsubstantiated assertions. Everything that is stated must be proven with either primary or secondary research, such as a quote from another peer reviewed article. This means that there is only one possible outcome that a reasonable person may come to after looking at the facts.

These critical thinking feminists feel like women are at a disadvantage writing under this set of rules and they basically want to be able to argue that yes 4=5=9 but it could also equal 7 depending on your point of view, your gender, your race or where you grew up.

[irnbru]

Identification and measurement of cross-cultural intelligence factors is a fashionable subject. The current arguments away from Spearman's classic general intelligence, or 'g' factor mostly revolve around the evaluation of the functional value of decision-making within a specific cultural context. Sternberg (2004) gives a good overview. This functionalism is exactly the type of drift toward 'math application' classes that are being suggested to replace algebra.

I subscribe to the Whorfian hypothesis perspective in that an individual is limited in their thinking to the concepts they can express in their language.

This is not a new idea and the old dude Confucius was saying that the correct path to wisdom was giving things their correct name.

If names be not correct, language is not in accordance with the truth of things.
If language be not in accordance with the truth of things, affairs cannot be carried on to success.
When affairs cannot be carried on to success, proprieties and music do not flourish.
When proprieties and music do not flourish, punishments will not be properly awarded.
When punishments are not properly awarded, the people do not know how to move hand or foot.
Therefore a superior man considers it necessary that the names he uses may be spoken appropriately, and also that what he speaks may be carried out appropriately. What the superior man requires is just that in his words there may be nothing incorrect

The above quote from an Analects dialogue which foreshadowed the social need to avoid Orwell's 1984-style doublethink newspeak.

Maths is a language and some ideas only mean anything and can be operated on or examined in the language of maths which is why Maxwell's field equations was brought up as an example in the earlier post. It also follows that I fully support your own position of achieving a proficient level of reading.

Regarding race and sex factors in standardised testing, there are the classic examples of self-identification of race as African-American before an SAT-analogue reducing points. Then the short-term increase following the Obama-effect. Likewise, if an Asian woman takes an SAT-M test, being prompted to identify by sex will reduce the test group's score, while identifying as Asian will increase the score. In sports psychology applied to golf, framing a putting test as a strategic activity will increase caucasian males scores and lower African-American scores, while framing putting as an athletic activity will do the reverse.

This all supports the position that perspective and self-evaluation limits the group, not intrinsic ability or the source material itself.

Elizabeth Spelke did a roundup of studies into math testing between males and females up to college graduate level. No real difference in actual cognitive capacity but a slight difference in problem-solving methodology between the sexes. One key point in the paper is that maths graduates are 50:50 male:female. One key point not in the paper is the comparative postgraduate achievements between the sexes.

Switching pure maths for applied (more practical) is exactly like reducing the critical thinking capacity and limiting the individual's capacity to operate on more complex concepts. It is dumbing down. The only valid argument I've seen for introducing more 'practical' problem-solving is that certain cultures are creatively limited and throttled in their capacity for divergent-thinking, i.e. issues with innovation in China. Remember that a rigorously-trained musician, artist, chess-player or sportsperson will have a much deeper vocabulary and set of tools to draw from.

Spelke, E. S. (2005). Sex differences in intrinsic aptitude for mathematics and science?: A critical review. American Psychology. 60(9), 950-958

Sternberg, R. J. (2004). Culture and intelligence. Am Psychol. 59(5):325-38