Mason's article - entrance slip

In his article, Mason looks at what types of questions are asked in a math classroom.  He argues that there is a certain way of asking questions which invites students to deepen their understanding of mathematics in an authentic way.  Asking questions in a way which does not presume that there is one, concrete answer which the teacher already has in mind, fosters a way of approaching mathematics which is more inquiry-based.  If students feel that the teacher is authentic when he/she asks for the students' help in solving a problem, they will feel more inclined to develop an original approach to arrive at the desired solution.  If the students don't feel constrained by the narrowness of a question, they may feel free to take a broader, more inquiry-based, approach to solving the problem.

During my long practicum, I must be careful not to give students very narrow questions which I already have pre-determined answers to.  I must try believe more in the students' ability for original and creative thought.  Hence, I must take an approach, like Mason's, to ask students broader questions, questions which allow for a multitude of answers, and questions which can ultimately foster the understanding of the students.

Micro-teaching exit slip - a reflection

Deeya, Ying Ting and I did a microteaching on the ambiguous case of the Sine Law.  Our presentation began with a brief summary of the Sine Law, followed by an inquiry-based lesson whereby students constructed triangles which show-cased the ambiguous cases of the Sine Law.
Overall, I think that we were weak when it came to time considerations.  Also, I thought that the presentation seemed quite disorganized because I do not think that we gave students enough information to be able to understand the inquiry-based part of our presentation - making triangles out of skewers.
Teaching inquiry-based lessons, I have learned, can be quite challenging, especially if time is short.  I felt that when we divided the class into three smaller groups, we lost the attention of a lot of the students, because they were no longer focused only on one presenter but on three.  If I were to teach this in a classroom, I would definitely spend more time showing the students what they are trying to discover, scaffolding their understanding so that they don't get confused later on.
We also lacked an appropriate hook.  Keeping students focused during the entire presentation requires that they be on-track right from the beginning.  Also, although this is not crucial, we lacked an appropriate conclusion because of time considerations.  We should have tried to include this regardless.

Micro-Teaching for Nov 30 - the Sine Law

Title: Learning the different cases of the Sine Law
Date: November 30, 2015
Grade Level: Pre-Calculus 11
Prescribed Learning Outcomes: B3. Solve problems, using the cosine law and sine law, including the ambiguous case.
General Purpose: This class will look at how the Sine Law can be used to find the unknown length of the side of a triangle given one angle and the lengths of the other two sides.  We will look at three cases: 1) where there is no solution, 2) where there is one solution, 2) where there are two solutions.
SWBAT: 3.5 Sketch a diagram and solve a problem, using the sine law. 3.6 Describe and explain situations in which a problem may have no solution, one solution or two solutions.
Probing of previous knowledge: We are assuming that in the class previous to the one we are teaching, students learned about the Sine Law and about how to prove it.  They also know basic properties of triangles, like how all the angles add up to 180 degrees.
Objective	Time	Activity	Materials
Introduction/hook	1 minute	Give students a little background information about the use of trigonometry - how it has been used early on as a way of measuring long distances, by calculating angles and smaller distances, and using perspective to approximate the longer distances.
Summary	2  minutes	Review the concept of the Sine Law (presented from a previous class).  Ask the students whether they think they can find an unknown side length of a triangle given the other two lengths and an angle.  Is this always possible?	·
Inquiry project	3x3 = 9 minutes	Illustrate the three different cases for Sine Law (whether one, two or no solutions) by dividing group in three, and having one instructor (Ying Ting, Deeya and Etienne) assigned to each group.  Each instructor will present a different case.  They will give students sets of three sticks, with which they have to make as many different triangles as they can.  Once they have come up with an answer, they have to explain why their solution is correct.  The instructor will help guide the students, writing the problem on the board and explaining why there is indeed only one, two, or no solution. The instructors will spend 3 minutes with each group, and then rotate, for a total of 9 minutes.	Measured sticks with which to make triangles.  Multiple sets for each case.  Three cases.
Handout+ summary	3 minute	Give students the handout on the Sine Law cases.  Go over the handout briefly.  Use it to show students that the three different cases necessarily have one, two or no solutions.  Illustrate this on the board, clearly.  Show the implications of this for when solving the Sine Law.
Summative Evaluation: By having students use the sticks to make up different triangles, students will illustrate their knowledge of the different number of triangles that can be made given certain lengths.

2-column math problem - work and thoughts

 

 

Hewitt Movie Class reflection - exit slip

I liked watching the movie today on a math teacher (Mr. Hewitt) not just talking about educating kids (as we have been doing for most of our BEd courses now), but on actually educating them.  I appreciated the techniques that he used - it allowed me to compare my own techniques to his, and to ask myself whether I would have done things the same or differently in his place.  I liked his use of repetition.  I also liked the way the he went around the class and made things more interactive for the kids.  I might have changed what he did slightly by writing more on the board the things I was saying.  But other than that, I thought that he demonstrated a good model for the way I would like to teach.

Hewitt Article - Entrance Slip

Those things which do not have to be the way they are, are called arbitrary.  Those things which must necessarily be as they are, are called "necessary".  The names we attribute to things, definitions we give to certain words, these are arbitrary.  A square does not have to be called a square.  It could be called a triangle.
There are others things, however, which cannot change.  The three angles of a triangle add up to a half turn - this is a fact which is not merely based upon convention, but upon reality.  The sum of the squares of two sides of a right triangle is equal to the sum of the square of the hypotenuse.  This also is true in every instance, no matter what name we give to certain terms.
In my math class, I would try to allow my students time to work out those things which are necessary.  Necessary things can be arrived at through intuition and through working out problems by oneself.  Those things which are arbitrary, that need to be told to students, I would try not to spend as much time on.  These things should be supplied as tools to the students, so that the students can proceed to work on those parts of mathematics which are not arbitrary, but necessary.

MOA Math Fair - exit slip

I just attended the Math Fair at the Museum of Anthropology. The students from West Point Academy were there, presenting mathematical problems in pairs of two.

I was very impressed by the presentations made by the young people. They were professional, and evidently very proud of their work. I was amazed by their eagerness in presenting their problems to us (the "grown-ups"). I felt they had put so much work into it, and were happy to show off the fruit of their labours. The students were cooperating very well together, and took turns presenting. They helped us with hints and clues as we tried solving the problems. They looked proud when we told them how well they did.

This experience of the Math Fair really encouraged me. From the way the students responded to this project, I would say that it was very successful. It would be worth doing in the schools.

Math Fair article

When I was in high school, we were encouraged to take part in math contests.  Each grade had its own contest.  I thoroughly enjoyed these contests, most of all because of the competition it engendered between myself and my peers.  We would always compare our results with one another when our test scores came out.

While I think that these types of math contests are positive, I think that the Math Fair approaches also has many positive qualities.  The fact that the Math Fair is not as competitive will certainly open it up to those students who may feel disuaded by the traditional math "contest".  I know that, in high school, I often felt like my performance on the math contests was inadequate - it was always the same 2 or 3 students who got the top scores.  A math contest would open up the possibility of all students having fun with math, not just the "best" students.  The Math Fair, I think, would encourage creativity and innovativeness, qualities which are really useful to develop at a young age.  The Math Fair would incorporate art (designing the poster), problem-solving, creativity, and oral presentation skills (for when the students present their problem).  For all these reasons, I would support running a Math Fair in my school.  I think that if I were to run such a fair, I would try to stress the communal aspect of it - I would encourage students to solve their problems by collaborating with others, and to present their problem in groups of 2 or 3, in order to cultivate delegation and inter-personal skills.

Calculus 12 - class review

Today, Pari and I looked at the curriculum for calculus 12 (taught in high school).  We looked at the content material and the way in which the curriculum was structured.

First of all, the content was similar to the one Peri and I had studied in our high school calculus classes.  The main focus of the material is on differentiation and integration.  However, there were some novel features.  The material included a section dealing with the historical foundation of calculus.  Students were expected to know about the different mathematicians who contributed substantially to the material presented.  Newton, DesCartes, Leibniz, and others, are studied.  This brings history into the context of mathematics.

Another feature of the curriculum was its emphasis on assessment of the material taught.  The curriculum provides criteria by which the students can self-assess their own or others' work.  The assessment criteria are specific to each section of the curriculum.  Students are expected to work out problems on their own or with others, and to develop innovative ways to both answer problems and to self-correct their own problem-solving skills.

Micro-teaching reflection - Greek Alphabet

I really enjoyed doing this micro-teaching.  I wanted to communicate to students a bit of my knowledge of Greek and particularly of the Greek alphabet.  Math students see Greek letters all the time in their classes, but usually do not get to learn about what these letters mean exactly.  As far as the lesson was concerned, the main thing I noticed was how quickly time went by.  I did not have the time to do everything that I wanted to do.  In fact, I didn't even come close.  The activity which I had planned to give students, which would involve them writing out the Greek letters on paper, did not materialize.  I found that the most efficient thing was just to write all the letters on the board.  Thus, in order to save time, I had to make the lesson less interactive.  If I were to do it again, I would try to focus on less content, and make it more interactive by providing enough time for an activity.  Most of the comments I received from my peers had to do we these two things: time management, and lack of interactive activity.  However, I did manage during my microteaching to ask lots of questions, which is usually what I do when I teach.  I love asking questions and probing students for answers.  I like using the blackboard a lot, and I think that for this micro-teaching, this proved beneficial.  I also trying to make the lesson interesting by giving a lot of practical knowledge to students - how to pronounce certain words and letters in Greek, and informing them that their former training was often incorrect (math teachers often don't know how to pronounce Greek!).  This made it possible for students to connect with my lesson and to draw applications for the future.

Battleground Schools - Entrance Slip

I found the history of the approaches to teaching mathematics to be very interesting.  I grew up in the 90s, which, according to the article, was a period of time when the NCTM Standards' approach to mathematics was prevalent.  I grew up with the mindset that objective standards to teaching math was the norm.  Course material has always been presented to me as an absolute reality which was pretty standard in all schools.  And as far as the content was concerned, I thought that the material taught to us was the only appropriate material for our age group.
In high school, I did get some newer perspectives, however, on the different ways of teaching mathematics, because students from Asian countries seemed to have had a better mathematical education (at least in terms of the level of difficulty or grade level).  However, I do not know how teachers in Asia differered in their pedagogical approach.
I find this article so interesting because it challenges my view that the way I was taught mathematics was the only way that had been tried up to then.  But in reality, pedagogical approaches to mathematics have not always been monoform, but diverse.  In fact, it is because of this diversity that correct pedagogy has become in some ways a political issue.
When I teach mathematics, I will certainly take a lot from the "new" ways of teaching - but I do not think that my way will be a totally novel way of teaching.  What I hope is simply to combine philosophy and thinking with mathematics, to reach people at many different levels, rather than at just the end-result level which has test-taking as its ultimate aim.

Lesson Plan for Mini-Lessons

Title: Greek Alphabet and Basic words
Date: October 21 2015
Grade Level: N/A
Prescribed Learning Outcomes: To teach students the 24-letter Greek Alphabet
General Purpose: This class will look at how to write out the letters of the Greek Alphabet, how to pronounce them, and a few basic words in Greek.
SWBAT: q Correctly pronounce all the letters in the Greek Alphabet. q Be able to read a few basic words in Greek.
Probing of previous knowledge: Students are not expected to have any formal training in Greek.  However, many of the Greek letters will be familiar to math students who have seen these letters used as variables.
Objective	Time	Activity	Materials
Introduction/hook	1 minute	Tell the students "good day" in Greek! Tell the students what I will be introducing to them.
Alphabet activity	5 minutes	With the students, go through the enter Greek Alphabet (all 24 letters) Write the letters on the board, and have the student write them out, while pronouncing them	·     paper and pencil
Explanation of friction	3 minutes	Write some basic Greek words on the board, and try to have the students read them out correctly. Note how close some Greek words come to English derivations of them!
Conclusion	1 minute	Present a Greek phrase to the students, taken from the inscription above Plato's Academy.  "Let no one who knows not geometry enter here!"  Read it out.
Summative Evaluation: By having all the students write out the 24 Greek letters, I know that all the students have thought about and applied their basic knowledge of Greek.  They can now present Greek letters with more confidence in a math classroom!
References: http://www.ibiblio.org/koine/greek/lessons/alphabet.html https://en.wikipedia.org/wiki/Greek_alphabet

Estimating the volume of a campbell soup water tank

First of all, I want to point out that pictures are deceiving.  Although it looks like the bottom of the bike wheel starts below the lowest point of the water tank, one can see on closer examination that part of the tank is buried in sand (understandably, to keep the tank from rolling away!)  Furthermore, since the top of the water tank is further away than the bike, we have to adjust our perspective slightly so as to take this into account.  Overall, I would estimate that the diameter of the water tank is 2.2 times the height of the bike.
To determine the height of Susan's bike, I measured the height of a medium-frame bike I have at home.  It was 102 cm.
The diameter of the water tank is therefore 102cm x 2.2 = 224cm.
According to an online source (https://sketchup-math.wikispaces.com/Campbell's+Soup), a Campbell soup can has a height of 4 inches and a radius of 1 and 5/16 inches.  Therefore it has a diameter of 2 and 5/8 inches.  Therefore, the height is 4/(2+5/8)=1.52 times the diameter.  Apply this now to the water tank.  The water tank's length (equivalent to a can's "height") will be 1.52x224cm=341cm.
The volume of the tank will therefore be pi*radius^2*height = pi*(224/2)^2*341 = 13450000 cm^3 = 13450000mL = 13 500L = 3550 gallons!

Imaginary letters from former students & reflection

Dear Mr. O'Toole,

This is Timmy.  I was a grade 10 student in your math class 10 years ago!  I was thinking about you recently and wanted to write a little note.  I don't know if you knew this, but after grade 10 I dropped out of the foundations of math to the essentials of math class.  I always found it difficult to keep up in class.  The rate at which you went through the material, though good for others, was always way too fast for me.  And then, there was the homework.  Oh, you assigned so much homework!  I could never get through more than half of it.  My parents could not afford a tutor for me, so I had no choice but to drop foundations math.  Though now, I sort of wished that I could have received some more help, so that universities would have looked better upon my report cards!

Timmy.


Dear Mr. O'Toole,

This is Amanda.  I was part of your grade 12 math class 10 years ago.  I just wanted to write to say how much I enjoyed math in your class, and how satisfying it was to be able to understand the material you gave us.  You always went at a good pace, with clear explanations, so that I was able to keep up with all the school work and even have time to be part of the math club.  You seemed to love math a lot, and communicated it in a passionate way.  Thank you for helping me to do well in class, which prepared me for later in the future.

Amanda

Reflection:
Perhaps my greatest struggle in teaching will have to do with the different levels of ability in math that I will encounter among students.  Some students will be stronger, others not so strong.  I think that I am going to have to find a way to communicate the material to all, while at the same time adapting it to certain learners through ways of teaching and through other strategies.  It will take experience, I think, to be very in-tune to the students' needs, and to learn different approaches which will help all the students in my class.  The issue about homework is a big one.  I will somehow have to try to keep my students accountable to the work that I give them.
This activity helped me to think from the perspective of the student, who does not always perceive things in the same way as the teacher does.  The teacher needs to be very understanding of every individual person.

Reflection on math art presentation - hyperboloids

I have never made a math-art project before.  I didn't know what to expect.  For me, mathematics throughout secondary school and university was about abstract concepts and theories.  So one can well imagine the benefit I gained from actually doing a hands-on math project.

First of all, there was the social dimension of the project.  Working in groups was fun, and was conducive to making new friends.  I think that in high school, doing a project like this (especially in earlier years when kids still don't know one another too well) would be very useful in building bonds and relationships between students, helping them to make friends.  Mathematics doesn't just have to be about formulae, it can also be fun.

As far as the hyperboloid structure is concerned, I learned a lot about mathematics.  I learned that a hyperboloid can be made out of straight lines, that these lines are parallel to one another, and that a line only intersects any other line (if they are not parallel) in one place.  I learned that one can take a two-dimensional lattice of skewers, and shape it in three-dimensions to form a hyperboloid.  The concept of a hyperboloid is therefore more comprehensible and is not outside my ability to understand.

I think that such a project can make certain areas of mathematics more comprehensible for students who have never though of mathematics outside the realm of paper and pencil.

Math That Matters reflection

This article was, I thought, very interesting.  It approaches mathematics from a social justice perspective.  This idea is not unlike ideas presented in other classes, whereby teachers might introduce social justice in all school classes.

My opinion regarding this matter might be somewhat controversial.  Following discussions I've had with students from other classes, I feel that teachers should not become political ideologues.  Although social justice certainly has a place in the curriculum, and ought to form the perspective of the teacher and of the students, I am wary about actually regimenting it so that it is the basis of the educational approach.  That is not to say, however, that I disagree with applying any of David Stocker's ideas in the classroom.  What I am hesitant to agree with, however, is when David Stocker speaks about applying these ideas in the classroom even to the disagreement of the parents.  I am a supporter of the idea that parents are the primary educators of their children, and I think that, despite the good intentions of teachers, a political approach to education would ultimately be too narrowing.  Children are more than political entities whose potential depends on how much they might change or influence society in the future.  As a teacher, I will be dedicated to forming individual students, not for the sake of political ideals, but for their sake.

Chinese plate-guest problem (without using algebra!)

If every two people share a dish of rice, then every person eats 1/2 a dish of rice.
If every three people share a dish of broth, then every person eats 1/3 a dish of broth.
If every four people share a dish of meat, then every person eats 1/4 a dish of meat.
Therefore, a single person eats the equivalent of 1/2 + 1/3 + 1/4 = 13/12 dishes of food.
Since there were 65 dishes in all, we can determine how many guests there were by dividing 65 with 13/12, which is the amount that each guest ate.
65/(13/12) = 60.  Hence, there are 60 guests.
This method only works if all the dishes of food were consumed entirely (that is, there were a perfect number of guests needed to consume all the food).
I think that cultural context certainly makes the question more understandable, and easier to relate to.  This problem presupposes an understanding that all the guests share the food equally, that they eat from common bowls, and that just enough food was put out to feed everyone.  In a western banquet, one would certainly not eat all the food, and just eat selections of the menu.  Underlying assumptions might sometimes be based on cultural context.

Reflection on my best/worst math teacher

The worst math teacher that I have ever had taught me during my grade 9 year in high school.  Objectively, she wasn't a bad teacher.  In fact, most students liked her.  She was our science and biology teacher.  But math 9 didn't require someone whose expertise was in mathematics.  She taught us math, and we were fine with it.  I have always done well in math, being at the top of my class, getting As.  But in grade 9, things were different.  I got a C+, primarily because I didn't try very hard to do well.  Math normally came second nature to me, so getting this grade surprised me a little.  Because my attitude towards school in grade 9 was very lax, I think that the teacher judged that I was not able to do exceptionally well in class.  And she treated me as such.  She never showed faith in my abilities, never gave me a chance to prove myself, and never encouraged me to do better.  Rather, she let me be, and I was fine with that.  However, I think it is a teacher's role to sometimes go beyond a person's comfort zone and to reach out to them.  Even if it's in math class.
The best teacher that I have ever had was my grade 11 math teacher.  He was funny, personable, interested in students' well-being, and had a real passion for math.  His degree was in math and music.  He told us math jokes, encouraged us to get involved in math contests, and was always friendly and good-humored.  He made us want to enjoy math because he enjoyed it himself.  And he gave us the opportunity to do the same.

Reflection on my TPI results

Compared to some of my classmates, whose graphs I have seen, my results are pretty even across the board.  However, this does not mean that there is nothing to say about these results.  First of all, my strongest category is that of transmission (TRANS) of the material.  This does not totally surprise me, for I certainly consider this to be one of the main purposes of teaching.  Do I think it is the primary purpose?  Well, yes and no.  I think that forming a student in a holistic way is done in large part through the process of teaching the material.  So, from that perspective, some of the points given to transmission may perhaps be considered to be being taken from the nurturing category.

I think that the holistic approach that I take to teaching is reflected by my results.  The fact that they are pretty even across the board indicates that I put about equal emphasis on every aspect of a student’s learning, and that when teaching I have multiple goals or purposes in mind, not just a single one.

One more observation that I made about my results is that my beliefs are normally stronger than my action.  I find this interesting.  It is true that I have strong beliefs about teaching and pedagogy, but when it comes to teaching, I do find myself to be quite passive as far as the effect that I may have on my students.  I take a hands-off, let-them-be-free-to-respond approach, which may or may not be for the best.  Experience and time will tell.

October 23rd

On October 23rd, I will be going to the Catalyst BCScTA Conference for Physics Student :)

How many squares are there in a 8x8 square chessboard?

Interesting question!  At first, one might suppose that the answer is obvious - 8x8 is 64, of course!  But the answer is not so simple.  One must ask, how many squares can be made in total - including the squares that can be made out of the other squares.

To answer this question, we definitely want to systematize our approach.  It would be way too complicated to just make random squares here and there, and to try to count them up on our fingers.  In mathematics, we often see patterns develop that can help us to determine complicated questions by first finding the solution to very simple ones.  In some ways, this is like the inductive approach to proving a mathematical statement or equation - first prove n=1, then assume n, and show that n+1 is true.  Let us try to solve this problem by considering the most basic case: a 1x1 square.  Obviously, there is only one way to form a square out of this.

Next, consider a 2x2 square.  In this case, we have the 4 individual squares, plus the one bigger one made up of the smaller ones.  So in total we have 4+1=5.

Then, consider a 3x3 square.  We have the 9 little squares.  Then, if we count the number of 2x2 squares, we find 4.  Finally, we have only 1 3x3 square.  So the total is 9+4+1.

Next, consider the 4x4 square.  We have 16 little squares, 9 2x2 squares, 4 3x3 squares, and 1 4x4 square, for a total of 16+9+4+1=30.

We definitely see a pattern emerging!  The pattern is as follows:

1x1

1x1+2x2

1x1+2x2+3x3

1x1+2x2+3x3+4x4

...

1x1+2x2+3x3+..........nxn, where n is the number of sides of the square.

This can be proven fairly simply by induction.

 

It is interesting - induction can be used to prove a whole series of problems having to do with geometric patterns.  Number theory and geometry (the most abstract and the most concrete topics in mathematics!) combine beautifully in order to determine the solution to these problems.  Wherever there are patterns, there is often a mathematical description to the pattern.

Instrumental vs. relational teaching - A reflection on today's class

Instrumental vs. relational teaching: a reflection of today's class discussion

Sept. 16, 2015

Today we looked at the pros and cons of teaching instrumental mathematics versus relational mathematics. Yet teaching doesn't have to incorporate just one or the other method. Both can be integrated. When teaching a subject, I think that both instrumental and relational should be taught together at all times. In philosophy, we would say that every object has both form and matter -- the form being the essence of the thing, and the matter embodying its accidental (secondary) attributes. I would propose that this analogy of form and matter fits very well with our understanding of relational and instrumental ways of teaching math. The structure (instrumental) is necessary as a scaffold in which to set the understanding of the student (relational). We teach the rules, the boundaries, the restrictions, the equations, while explaining how these equations and rules are derived, and how they reflect and can predict reality. To give but one example, consider the Pythagorean theorem. For a student to understand this theorem, he must first have an understanding of what a right triangle is. The Pythagorean theorem provides us with an equation which relates the relationship between the different sides of the triangle. Once the equation is memorized, and the framework is understood, we can then explore the consequences of this equation by noticing that the hypotenuse is always going to be larger than the other two sides. We can consider one or two different triangles, which we can draw out accurately on the board, and test our equation on them. We can consider extreme cases where one of the sides is almost zero in length. We can show how if we double the two right-angled sides of the triangle, then the hypotenuse is also doubled in length. We can also prove the Pythagorean theorem using the simple visual proof of triangles inscribed in a square. The understanding will follow within the framework of the instrumental equations and rules.

I think too that if the teacher has a very relational understanding of mathematics, then he will communicate his way of understanding to the students. I have noticed this whenever I taught high school math -- I would always investigate the why of what I was teaching. And yet a prudent teacher must know where the middle road is, and how much time ought to be spent simply on imparting meaning vs. intrumental teaching.

Skemp's article reflection

Relational Understanding and Instrumental Understanding

A reflection on Skemp's article

The first thing the author wrote that struck me was his differentiating the two ways in which students learn mathematics.  He maintained that there is an essential difference between the relational and instrumental approaches to mathematics.   I concede that these are definitely two different approaches to the way we study mathematics, but I never considered them separate entities altogether.  The author was successful, I think, in demonstrating that the instrumental approach to mathematics can be very limiting, and that it really should not even be called “mathematics” in the same way as is understood when we talk about relational mathematics.  The second thing that struck me was the author’s summary of the pros and cons of both approaches.  First, he talks about how “relational” mathematics is superior to “instrumental” mathematics.  However, there are some serious limitations in teaching mathematics relationally.  For example, this way of teaching might take a lot longer, and therefore might be impractical.  It made me wonder, where do we draw the line in terms of how much time should be sacrificed just so that we can teach in this way?  Finally, what struck me was the way in which the author radically separated the teaching of instrumental mathematics from the teaching of relational mathematics.  He seems to have a very black and white view.  I think that we have to consider these things more as a spectrum – our teaching will involve both instrumental and relational approaches.  We cannot teach everything intuitively, nor can we teach students only as though they were robots.  I think what the author failed to do was to see the gray area in which we can get the benefits from both ways of teaching math.

This is my new blog