Friday, January 20, 2012

[MathEd] Proposed New Framework for Mathematics Education

Figure 1 – My Proposed Framework

     In this article, I propose a framework for mathematics education that can be used in curriculum (re-)design, lesson design, evaluation of the attained curriculum, as well as analysing mathematics education or educational technology initiatives.  As a citizen of Singapore, I do hope that at least some of my ideas (in their original spirit) gets considered and adopted in my own country, but I want to share this with everybody in the world –  whoever cares to listen and engage.  I hope to spark a global conversation among students, parents, teachers, industry leaders and educational leaders regarding the future of mathematics education, according to needs and challenges that are currently felt in the 21st Century world and as well as unforeseen needs.

     When we talk about mathematics curricula, we need to distinguish between the intended curriculum (what we think should be taught), implemented curriculum (what teachers actually teach) and the attained curriculum (what students actually end up learning).  These are different things.  My framework attempts to build upon the strengths of the present Singapore (intended) mathematics curriculum, and is influenced by my post-graduate studies of the academic literature in mathematics education, as well as
my observations and reflections of my personal experiences in my career as a teacher, tutor, instructional designer, educational software developer, researcher and consultant.  To explain my framework fully, it would probably take many pages and chapters.  Here, I shall give an introduction to my main ideas.

Crisis in Mathematics Education – “Iceberg” Metaphor
Figure 2 – What most schools (try to) focus on for maths
     Mathematics is an enterprise of gaining knowledge about the regularities and patterns
in our universe that has been practiced by people from different cultures in history throughout the world.  Most schools around the world try to teach only certain mathematical facts and procedures, which are only a very small part of what mathematics is really about (hence “the tip of the iceberg”).  And even with this, they are already struggling.  Politicians and curriculum developers tend to focus on tests and examinations that assess concepts, skills and processes (algorithms or methods of calculation).  Parents naturally want their children to “do well” in mathematics.  Many people think that the best and “objective” way to indicate this is via paper-and-pen examination and test grades.  For the sake of “accountability”, most teachers around the world seem to be pressured to take an exam-oriented approach to teaching mathematics (and much else).  What tends to get ignored are things like the development of students’ ability to solve real-world problems, the ability to learn on their own, the willingness to engage in life-long learning, the ability to “figure out things” on their own, a love and thirst for knowledge and sense-making of the world, the cultivation of values (e.g. appreciation of beauty, connection with other disciplines, precision, rigour) and dispositions (e.g. creativity, patience, meticulousness, succinctness, critical thinking, a questioning mind … etc).  The “mathematics” most students get at the end of their school career is probably some spotty recollection of a few concepts and a few tricks and this fades in their adult life.
Figure 3 – What most students achieve in maths

     Furthermore, educators around the world generally fail to connect with students’ identities: a sense of who they are as human beings in this world, their roles, goals, wishes, aspiration, ambitions, decisions and life-stories, and how mathematics is relevant to the development of these matters.  This is true even of the best students, what more of the rest of the students.  Students who are “good” in mathematics (in our current education systems) may not like mathematics or see its relevance to their lives.  They may just be able to grudgingly attain good “performances” in mathematical tasks.  The bulk of the students are disengaged.  They think “You know … maths is just one of those things we have to get through in order to get to the university course of my choice”.  The “worst” students hate mathematics and the system, and they become disruptive and anti-social – they literally “deconstruct” the system and yet they do not have anything constructive to show for in its place.

Figure 4 – What “mathematics” computers can already do now
     We now live in the Information Age where information technologies (e.g. graphing calculators, Computer Algebra Systems, Wolfram Alpha, … etc) are getting more and more advanced by the day and can now do many of the numerical and algebraic/symbolic calculations that schools are trying so hard to teach to human beings.  [2015 Update: Recently, mobile apps like PhotoMath appeared on the market.  These apps allow students to snap photos of mathematics homework problems and the apps will solve the maths problems for them, including the working. ]  Actually, we do not need human beings to do the procedures of algebra and calculus anymore – machines can do them much faster and with less hassle.  There is no need to have them sit for mathematics classes to be taught with boring lectures and colourful textbooks, occasionally spiced up by some “math apps”, and then fail to learn perfectly.  Dear reader, if you have not realised it by now, this spells
                                              D I S A S T E R. 
The human beings who graduate from our mathematics education (if ever they do) are redundant!  It is a false comfort that today we have technology that can even do mathematics homework for students.  In fact, this is the very reason that these students are irrelevant in the current and future job market.  Furthermore, they do not acquire a wholistic mathematical education for their adult living.

Figure 5 – What humans need but do not learn in most  schools

     My proposed new framework attempts to address these challenges by putting emphasis on the deeper things.  This is not just about economic survival, but it is about what is most important for us as human beings trying to make sense of this universe as we live in it.

The Context

     In my framework, the learning of mathematics takes place in the context of real-life (symbolised by the land and sky) and a community (represented by the ocean).

     The “Iceberg” points to real-life: that means students link mathematics to contextualised real-life applications and authentic problems.  Students see how mathematics is relevant to their own lives and how mathematics is being applied.  [This does not mean sacrificing generalization and abstraction, but being able to see how the processes of generalisation and abstraction, when done properly, can be transferred to other contexts or new contexts.  This also means connection with other disciplines or subjects.]

     The community refers to fellow learners (not necessarily form the same age or country) and teachers and experts.  Instead of competitive individual learning, collaboration and connection with the world beyond classroom walls, contribution to society is encouraged.

The First Five Layers

     The First Five Layers of my “Iceberg” framework cover the following:-
     (1)  Concepts
            § Numerical     § Algebraic       § Geometrical
            § Statistical      § Probabilistic   § Analytical
     (2)  Skills
            § Numerical calculation   § Algebraic manipulation   § Spatial visualization
            § Data analysis   § Measurement   § Use of mathematical tools   § Estimation
     (3)  Processes
            § Reasoning   § Communication and connections
            § Thinking skills and heuristics   § Application and modelling
     (4)  Metacognition
            § Monitoring of one’s own thinking   § Self-regulation of learning
     (5)  Attitudes
            § Beliefs   § Interest   § Appreciation   § Confidence   § Perseverance

The Deep Layer

     Below these five layers, we have the following:-
     (1) Problem Solving
           § Understanding   § Planning   § Executing   § Evaluating   § Reflecting
     (2) Dispositions
           § Habits of Mind   § Transfer of Learning
     (3) Values
           § Purpose of Learning   § Utility   § Aesthetics
     (4) Epistemology
          § Ways
of knowing  § Logical reasoning  § Plausibility and number sense
          § Life-long Learning   § Self-Directed Learning   § Critical Thinking

     The above are very important, but these are just aspects surrounding identity
     (§ Character-building   § Roles   § Life-story   § Being and becoming)

What these all mean

     What these all means is in my conception of an ideal student who graduated under this mathematical education framework, this person is someone who has a strong sense of who he/she is in this world and what he/she wants to do with his/her life (identity).  As part of this core identity, he/she is able to solve problems, has desirable mathematical dispositions, values, is a life-long self-learner who knows how to figure things out on his/her own.  The mathematical concepts, skills (including the appropriate use of technology), thinking processes, metacognition and attitudes are built upon this core.  This person is able to collaborate with other people in real-life (including the ability to connect with other subject disciplines).

Connection with other disciplines/subjects

     I have alluded to connection with other disciplines/subjects.  What I have said regarding the crisis in mathematics education is probably largely true in disciplines/subjects.  One can imagine that other disciplines/subjects (e.g. biology, physics, literature, history, geography … etc) all have their similar “icebergs”.  Actually all these other icebergs are connected at the deep level, with “identity” as the common core.  Human knowledge has traditionally been dissected and put into silos for different disciplines for ease of handling, but in reality, all aspects of knowledge and learning are interconnected and there are no artificial boundaries.

Questions to Ponder

1.  Do you think my framework is practical?  Do you know of any places and/or
     schools already implementing all aspects of my framework (without necessarily
     putting them in the format that I have described)?  Which school?
     Which district / province / country?

2.  How would you redesign your province’s mathematics curriculum?

3.  If you are a current school teacher, would you want to redesign your next mathematics
     lesson after reading this article?

4.  Using my framework, how would you approach the evaluation of the attained
     curriculum (what students end up learning) in your country / school / district?

     Remember: students do not just learn facts (e.g. “1+3=4” ) and skills (e.g. factorisation)
     They also learn knowingly or unwittingly attitudes (e.g. that “mathematics is boring”,
     “it has nothing to do with my life”, “Oh!  It’s just a bunch of calculations”, “it has got
     nothing to do with logic”, “answers are what matters, not how you got it”, “just learn it
     from the teacher”, “don’t give me that cr** about reasoning, just give me the facts”

5.  Using my framework, how would you evaluate your country / school / district’s
     implementation of your curriculum?  Do you see any gaps in the way teachers actually
     teach your curriculum?  What are you going to do about it?

6.  Do you agree with everything I have said?  Do you have anything to add or take away?

7.  How does your school’s technology use fit into this framework?  How does it, for
     example, support students’ mathematical epistemologies (i.e. they way they learn, and
     the way they critically assess the knowledge that they have acquired via searching,
     experimentation, … etc)?

8.  Consider Apple’s latest initiative to put cheaper-than-paper-textbooks material on the
     market.  If you were to use a red-coloured pencil to shade the areas being covered in
     my framework, what areas would be shaded?

9.  Any other business …


     Actually, there is no conclusion.  We have only just begun.  With my introduction, I hope everybody has a clear idea of the issues we face today and what areas need to be addressed.  You may agree or disagree with me, or you may want to suggest some things.  Let the conversation begin.  Put your comments/feedback below or email me.


  1. 1. I totally agree with your framework. The challenge is how to implement it in a way that is encouraging to the students. Students nowadays are overloaded with tons of assignments, with very little time for sports or play. I know this teaches time management skills but I'm sure there are better ways to do it. I used to participate in my schools' music/sports teams simply because 'points' were awarded. It seems utilitarian but it really does build character. In fact, I feel that more incentive should be given for all schools to encourage the students to take up such extra-curricular activities. It is the first step in giving the students a sense of pride that they have some skill that is not part of the run-of-the-mill classes.

    2. I'm not sure how it is now but I feel that mathematics should be taught from the ground up, or from the bottom of the iceberg in your metaphor. I did have the privilege of such an education I think.

    5. Some of my teachers gave photocopied handouts, some wanted us to copy down everything by hand, some were in the middle ground, leaving some gaps in our handouts. I personally liked complete handouts but maybe the teachers in the middle ground were helpful too. Arguably, the teachers who wanted us to copy down everything were teaching us to write, listen, and think of different things at the same time.

    7. I went to school in the age of information scarcity. Now we are living in the age of information abundance. In my school days, we used to hoard information, because knowledge is power. This probably taught us teamwork. ^_^ Now information about everything is available on Google and Wikipedia. We can get the answer in seconds. The dark side is that it is much more difficult to discern the true facts from the false facts. This is why I feel that (physical) schools are still highly relevant, because the teachers have a moral responsibility to convey the true facts. With Facebook, information spreads like wildfire. I sincerely hope that students can develop the will power to log off their virtual life and take part in real physical activities.

    8. This is Apple's marketing strategy. A lot of giant companies use this strategy as well, especially in software markets. Powerful software is made cheaply or freely available to schools to encourage students to learn them. After they leave school, when they see an applicable task, that same software comes to mind. The same goes for manufacturer lock-in for hardware. It makes some business sense to entwine the customer so much in their hardware that they won't or can't go for another company's products. The Data Liberation Front has taken the revolutionary course of allowing export of personal information from Google services. Certainly laudable, but there aren't many good places to send our information to anyway. My feeling is that the more companies try to lock in their customers, the more they want to get out.

  2. thanks for your reflections. if this thing is really implemented, we'd probably need to overhaul the exams / assessment system. sorry, i didn't have time to explain everything yet. i write only when i have spare time. :-)

  3. I am interested in the very top of the pyramid. What I see happening in the schools (I am a substitute teacher, so I am in a LOT of schools) is that teachers attempt to teach the concepts, but this isn't very successful, so, instead, they really focus simply on teaching the skills. The good students learn the skills well (and the poor students, poorly, or not at all), but they don't learn much about the concepts, because the beauty of the concepts and their relationships with other concepts is hardly touched on at all.

    On the other hand, I am not convinced that everyone needs to understand the real mathematics behind the concepts. Most people do need the some simple concepts and simple skills. Whether they need to understand the mathematics or simply used it is a question I am still debating with myself.

    1. thanks for the comment. two days ago, i was just tutoring a student (from a top Singapore school) whose teacher set 8 homework questions similar to
      sketch the graph of |x^2 - 2x - 1| for -1 < x < 4
      i was wondering: why did the teacher set so many similar questions to torture the student. [yes, Singapore students (are made to) work very hard.]

      guess what? good news! Wolfram Alpha can do the job for the student
      see here

      the bad news: when the student grows up, he/she would find it hard to get a job if Wolfram Alpha can do it. so why are educators still focusing on teaching and testing the top of the pyramid? note: i'm not saying the top of the pyramid is not important. mathematics has a long history of how knowledge is discovered and difficulties conquered, of interesting personalities (men and women mathematicians) ... and a lot of culture ... students are missing out on all these, even in Singapore. our current students need to be able to discover, research, think critically, solve problems using maths (and other disciplines) to address issues of the day e.g. energy, environment, over-population, ... etc. and of course, as part of their jobs they can use calculators and software (like Wolfram Alpha, MS Excel, ... etc.) to help them. Hopefully, computers won't be advanced enough to do human language processing and critical thinking, otherwise I fear for our human race. Nevertheless, some big companies are trying ...

  4. Oh, I probably should add that I am in the United States. (and that last sentence above should read, "...or simply use it..."

  5. Seymour Papert has an appropriate quote, shared today on the Daily Papert, which may answer your question Laura.

    “Being a mathematician is no more definable as “knowing” a set of of mathematical facts than being a poet is definable as knowing a set of linguistic facts. Some modern math ed reformers will give this statement a too easy assent with the comment: Yes, they must understand, not merely know. But this misses the capital point that being a mathematician, again like being a poet, or a composer or an engineer, means doing, rather than knowing or understanding.” Seymour Papert, 1970

    In other words, if we want students to learn how to think like a mathematician, they have to do what a mathematician does, which is to construct ideas from existing ideas, to do mathematics. Learning skills or concepts in itself should not be the primary goal - learning ways of thinking should be.

    1. I agree!

      Unfortunately, most schools in most countries (except possibly Finland) still test and teach according to specific skills and content. Besides Curriculum Design, viable Alternative Assessments seems to be a big issue.

      If I had my way, I'd probably give the students an "unseen" pure or applied maths problem, lock them in a room for 3 days, but provide all the reference books, Internet access and pizza ... see what they can come up with by deriving their own formulas and/or solving the problem.

      Another idea I have is to get the teacher plus one or two external referees to follow the students through their mathematics learning journey to see to what extent each student has manifested or improved upon a set of intellectual dispositions (e.g. rigour, creativity, perseverance, pattern seeking, ... etc.). acquistion of skills and content becomes just a by-product of these. is this idea too expensive to implement?

      what do you think?

  6. I would look at John Mighton's JUMP Math (non-profit out of Toronto Canada) that engages students' in conceptual and procedural mastery and importantly identity and relevance. The third party studies done on his work especially in underprivileged schools are amazing. His philosophy and methodology map very well to this framework. Especially interesting is his book "The End Of Ignorance".