A process of learning, unlearning and relearning

The interesting thing is that strategies that fail in the long term are precisely the ones  that we used to teach these new concepts in the short term. we relate numbers to counting, often using real “counters”. We relate multiplication to repeated addition. There is nothing wrong with this. Mathematics builds new ideas on old ones. it is hard to see hoe else one can teach it but eventually the training wheels have to come off the bike: students have to internalize the new idea. the art of mathematics involves switching effortlessly from one of those viewpoints to the other, do however, be aware of it when teaching. if one of the students  is having a trouble with a procept (process cum concept), that cause may be a past failure to proceptualise one of the processes involved. thus the job of a teacher is to back track through the series of ideas that leads up to the new one – not so much so on looking for the first place where the student fails to answer a question but rather looking for the first place where they can answer it  only by using some simpler idea as a crutch.

The 4 critical questions to follow when planning activities for children:

– WHAT do you want the child to learn? (i.e., the objectives)

– HOW do you know if they have learned? (i.e., how to assess or to see if a child has learned)

– WHAT IF the child is unable to do so? (e.g., a struggling learner)

– WHAT IF the child is able to do more? (e.g., an advance learner)

Likewise, whether I am delivering a lesson or talking with my students, I have to remember that what seems perfectly obvious and transparent to me may be mysterious and opaque to those who have not encountered the ideas before. There is room for input from the teacher myself, and there is a delicate balance between helping the students by putting my own stamp on the material, and confusing them by introducing too many extraneous ideas or strategies.

I see the need in building spaces where each person is visible to me and everyone else – and most importantly to themselves. I believe students should sense their own unique power and potential where in my classroom, each is known and understood, recognised and valued; asking questions out of the world, to interrogate common sense, to challenge the orthodox ‘why?’; so much so that the classroom is unsettled in such a way that students have a sense of curiosity and wonder and astonishment.

The value of student talk

Lewis & Tsuchida (1998) quoted a Japanese teacher as saying, “A lesson is like a swiftly flowing river; when you’re teaching you must make judgments instantly” (p. 15). Recognizing shared student thinking as a teachable moment is one of the instantaneous judgments that are made during lessons. Recognizing the pedagogical and mathematical value in students’ thinking in the moment is a difficult step in the process of using students’ thinking, even for experienced teachers (Chamberlin, 2005).


The teacher was teaching mathematics in a class.

“in order to subtract, things have to be in the same denomination. for instance, we couldn’t take two apple from three oranges, or five dogs from 6 cats. do you understand?”

the class seem to understood until one little boy asked, “But teacher,” he said, “how come we can take four quarts if milk from three cows?”


Learning how to orchestrate an effective classroom discussion is quite complex and requires attention to various elements.  The goal of which is to keep the cognitive demand high which students are learning and formalizing mathematical concepts, not for them to tell their answers and get a validation from the teacher

As students describe and evaluate solutions to tasks, share approaches, and make conjectures, learning occurs in ways that are otherwise unlikely to take place. Similarly, students come to see the varied approaches in how mathematics can be solved and see mathematics as something they can do.


math joke

The value of student talk cannot be overemphasized.


The art of writing is a reflective process

“Language, whether used to express ideas or to receive them, is a very powerful tool and should be used to foster the learning of mathematics. Communicating about mathematical ideas is a way for students to articulate, clarify, organize, and consolidate their thinking. Students, like adults, exchange thoughts and ideas in many ways—orally; with gestures; and with pictures, objects, and symbols. By listening carefully to others, students can become aware of alternative perspectives and strategies. By writing and talking with others, they learn to use more-precise mathematical language and, gradually, conventional symbols to express their mathematical ideas. Communication makes mathematical thinking observable and therefore facilitates further development of that thought. It encourages students to reflect on their own knowledge and their own ways of solving problems. Throughout the early years, students should have daily opportunities to talk and write about mathematics.” – Although the final representation of a mathematical pursuit looks very different from the final product of a writing effort, the mental journey is, at its base, the same – making sense of an idea and presenting it effectively.

When children make connections between the real world and mathematical concepts, mathematics becomes relevant to them. As mathematics becomes relevant, students become more motivated to learn and more interested in the learning process. Students continually want to know the relevance of math to their everyday lives and journals can give them this insight – spending more focused time thinking about the ideas involved  as students make an effort to explain their thinking and defend their answers. If students understand the importance of mathematics and how it relates to their own lives, they will be more likely to engage in classroom activities.

Writing can assist math instruction in two ways – by helping children make sense of mathematics and by helping teachers understand what children are learning. 

Mathematics in the atmosphere

It is unwise to permit students to study place-value concepts without encouraging them to see numbers in the world around them. Neither do we need a prescribed activity to bring real numbers into the classroom. As children get a bit more skilled, the interest in numbers can expand beyond the school and classroom. All sorts of things can and should be measured to create graphs, draw inferences, and make connections. Collecting data and grouping them into tens and hundreds (or thousands) will help cement the value of grouping to count and compare. The particular way you bring number and the real world together in your class is up to you. But do not underestimate the value of connecting the real world to the classroom. Take too “practically minded” an attitude and you stifle true creativity, to everyone’s detriment.


Teach not the way you are taught (conventionally) but they way you are trained (to be unconventional)

The Way We Used To Multiply

The old way to multiply required a student to add the products of 36 x 4 and 36 x 2. The trick is to add that 0 at the end of the second product.

The Way We Used To Multiply

How Kids Learn To Multiply Now

These days, students add four products to get the answer.

How Kids Learn To Multiply Now

You cannot become good at algebra without a mastery of arithmetic but arithmetic itself is no longer the ultimate goal. Thus the emphasis in teaching mathematics today is on getting people to be sophisticated, algebraic thinkers.

That doesn’t mean that kids can skip learning their multiplications tables. But the way it’s taught now is you get to the multiplication tables by understanding the number system and understanding what numbers mean.

If it sounds too complex for adults, let alone 7-year-olds.  With all due respect, there’s nothing elementary about elementary mathematics education.

Consider, then, a teacher who tells her students what a “ratio” is, expecting them to remember the definition.  Now imagine a teacher who has first graders figure out how many plastic links placed on one side of a balance are equivalent to one metal washer on the other side.  Then, after discovering that the same number of links must be added again to balance an additional washer, the children come to make sense of the concept of ratio for themselves.  Which approach do you suppose will lead to a deeper understanding?

To my mind, the most important feature of good teachers is that they put themselves in the student’s position. Rather than giving clear and accurate instructions and grading tests; the main objective is to help the students understand the material. Whether one is delivering a lesson or talking with one’s students, one has to remember that what seems perfectly obvious and transparent to one may be mysterious and opaque to those who have not encountered the ideas before. There is room for input from the teacher myself, and there is a delicate balance between helping the students by putting my own stamp on the material, and confusing them by introducing too many extraneous ideas or strategies.

To say that teaching from a constructivist perspective is characterized by a paradox – don’t give young children more than they can handle, but do give them a chance to show you what they can do – is to put this positively.  The flip side is that the Old School manages to screw up on both counts, simultaneously failing to understand children’s developmental limitations (“Just drill ‘em until they get it”) and failing to appreciate their minds (“Use the technique I showed you”).  This double fault reflects on educators who “overestimate children academically and underestimate them intellectually.” Unconventional teachers dedicate themselves to avoiding both traps.

Acceleration is not Enrichment

The distinction of them both is such that:

“Acceleration” refers to any strategy whereby individual pupils, or groups of more able pupils, are systematically fed standard curriculum work months or years ahead of their peers, thereby putting their learning (permanently) out of phase with that of their peer group


Enrichment”  refers to any strategy which seeks to serve the needs of able pupils in ways to avoid the above – to provide extension work which enriches the curriculum by requiring deeper understanding of standard material.


Enrichment is a term often misused, rather it is Acceleration mostly in operation for most eyes are caked shut with a dust of deception. It is undeniable that acceleration  has a strong appeal to most people (as with stakeholders) who are under pressure to trumpet their achievements infused with some sense of “kiasuism” Not to mention, society applauds for those who pursue an education to prepare for a career. My personal take is we should believe that children are significant contributors  rather than just objects or passive recipients of our activities paved for them – to equip them with the skills of being a learner so that they may inherit the ‘earth’ in times of change, rather then preparing them for the economy.


“The joy of confronting a novel situation and trying to make sense of it – the joy of banging your head against a mathematical wall, and then discovering that there may be ways of either going around or over that wall”
Page 43 Schoenfeld (1994)


Pre-course Reading Chapter 1 and 2

“Education is the point at which we decide whether we love our children enough not to expel them from our world and leave them to their own devices, not to strike from their hands their chance of undertaking something new—but to prepare them in advance for the task of renewing a common world.”

-Hannah Arendt

“Suppose there were 20 sheep in a field, and 10 of them jumped a fence; how many sheep would be left?”

“None,” called out little Billy. “Billy, I’m surprised at your answer,” replied the teacher.

“Surely your arithmetic is better than that.”

“You may know arithmetic, teacher,” replied Billy, “but you don’t know sheep. If one jumped, they’d all jumped the fence.”

Creative logic cannot be easily refuted. If you think outside the box, you be surprised at your own creative genius. Creativity can pave a way for breakthrough to problems. ‘Since there is nothing new under the sun, creativity means simply putting old things together in a fresh way.’ (Sherwood E. Wirt)

Today’s math curriculum is teaching students to expect — and excel at — paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. I love math but to be a wonderful teacher of mathematics, to inspire the uninspired, requires me to confront my personal beliefs on what it is to do mathematics, how I go about learning mathematics, how to teach mathematics through reasoning  and sense making, and what it means to assess mathematics so that it leads to targeted instructional change. The change, then, has to begin with the teacher (A teacher affects eternity – he/she can never tell where his/her influence stops), to model much joy to be had in solving mathematical problems and nurture that passion in my students.

Teaching is the vocations of vocations, because to choose teaching is to choose to enable the choice of others. It is about the business of empowerment, the business of enabling others to do well -an act of hope for a better future. The rewards of teaching are neither ostentatious nor obvious – they are often internal, invisible, and of the moment. But paradoxically, they can be deeper, more lasting and less illusory. There is a particularly powerful satisfaction in caring a time of carelessness and of thinking oneself in a time of thoughtlessness. The reward of teaching is knowing that my life can still make a difference. As long as I live, I am under construction, becoming a teacher, learning to teach, practicing the art and craft of teaching. I am still trying to achieve wonderfulness.

The art of intellectual challenge of teaching mathematics involves becoming a student of my students, unlocking the wisdom in the room, and joining together a journey of discovery and surprise. The ethical demand is to see each student as a 3-dimensional creature, much like myself, and an unshakable faith in the irreducible and incalculable value of every human  being.

Anyone can learn mathematics for its ability is not inherited. Therefore, we should not be intimidated by math, because we’re slowly redefining what math is.


In this video “Math Needs A Makeover”below, Dan Meyer asks, “How can we design the ideal learning experience for students?” Meyer has spun off his enlightening message — that teachers “be less helpful” and push their students to formulate the steps to solve math problems — into a nationwide tour-of-duty on the speaking circuit.

“I teach high school math. I sell a product to a market that doesn’t want it but is forced by law to buy it.”

Dan Meyer  



creative minds

the love for math