Interview Questions to Students and Teachers

Questions to students:

1.   Name three things that will greatly improve your math learning.  Explain why.
2.   Is math among your most or least favorite subjects?  Explain further. (Here, the student many be comparing the characteristics of other subjects to math)
3.   Apart from being a university requirement, is a good math education important?  (Is math relevant in real-life?)
4.   Does the use of technology (graphing calculators, smartboards, virtual manipulatives) enhance your learning?  Why or why not?

Questions to teachers:

1.   Name three things that you find most challenging about teaching mathematics to secondary students.
2.   How do you motivate disinterested, underperforming, don't-care-if-I-pass students to love math?
3.   What for you is authentic assessment?  How do you practice this?
4.   How do you balance the demands of the curriculum with teaching for understanding?
5.   How do you ensure judicious use of technology to maximize learning?

Mr. Not-so-great Fictional letter

Hi Mr. Not-so-great teacher,
Just a short hi from a former Grade 10 Math student who is now a Math teacher in Bahrain.  I bet you still remember me as I my always spell trouble- homework/seatwork.worksheest are either incomplete, obviously copied hurriedly, or non-existent for all I care.  It's amazing how I can come up with creative ways to tune out of your class for almost the entire year, while managing a semblance of deep thought or profound reflection. I've honed my astral travel skills that year, my thoughts in some far-flung exotic resort, while my body pinned down in my  desk.  I have not learned an iota of useful information from you, with all the mumbo-jumbo about projectile paths, exponential decay or inconsistent systems.  To be fair, I've not learned anything at all from my other teachers as well.  One thing I can credit you though, you generally leave me in peace.Perhaps you've divined the futility of any effort.  Oh, and that brings me to why I'm writing after all these years.  For some odd reason, my Grade 10 report card survived the Great Bonfire.  That was when I burned all my school stuff in a nice bonfire when I freaked out in the early part of Grade 12 coz I was so pressured to graduate that year.
Except my Grade 10 card, which is now right in front of me. At the end of the term, you said and I quote: Bryan has a lot of potential in excelling in math as he is can be focused if he wants to. I don't know if that potential stuff lay buried in my unconscious mind  all this time.  What I know is that I'm now a math teacher, and a pretty good one at that.  So what are the odds of that, eh?
Incorrigible student


Greatest hopes: That my students will appreciate the inherent beauty of Math and gain profound satisfaction in doing math the rest of their lives.

Greatest fears:  That despite best intentions, my Math class was so traumatic or boring or irrelevant that a student would want to erase all memories of it in her long-term memory.

Fictional letter

Dear Mr. Favorite Teacher,
Hello, still remember me from your Math 12 Foundations & Pre-Calc class?  Perhaps not,  with so many students milling around you, asking endless questions, seeking help in your assigned homework or even former students asking math questions assigned by OTHER teachers!  You seem not to have time for yourself in school, and even after school as you go online with the class blog every single night, responding to our endless queries, comments, complaints about Math in particular and life in general.  I know you once said that you're a face person and not a name person so I"m sending you a before (was it really 10 years ago?) and after (with my husband & 5-year old daughter) picture of me.  (Download attached file first before proceeding any further).
Ring any bells?  I can almost hear your gasp and hear your OMG!  Yes I'm the quiet student always sitting in the back row where I prefer to work alone.  But unlike other teachers, you always found a way to reach me and check on me every single class!  I really like it when you give us time to work on our own worksheets and you move about checking on ALL students, and commenting on how well we're progressing in our worksheets.  I'm eternally afraid to volunteer my answers in class, moreso, to ask questions if something is not clear.  Looking down on my work, you seem to know exactly what part of the process I'm having trouble with, and nudge me gently towards the correct path.    For that, I'm really grateful as it took the stress of learning math.


 

Summary: Battleground Schools

             The article describes two equally strong but polarized views of math education in 20th century North America, particularly the US.   The prevailing view at any given point in history depends on the current political and economic state of the US and the global scenario.  Hence the terrors of fascism in the 1940s together with  increasing migration, urbanization and industrialization,  rendered the static, rigid curriculum  wholly inadequate, spawned a distrust for the infallible teacher and called for a paradigm shift to a  more meaningful, practical education by "doing math."  John Dewey figured prominently in the progressivist reform of early to mid-20th c.

           Then a second major shift came in the post-war 60s when the US dominance in the global arena is threatened by Russia's seeming superior technological capabilities.  Heavily influenced by the ultra-conservative, highly elitist Bourbaki group, math sentiment swung to the conservative end, where abstraction reign, and university-level math is injected in K-12 curriculum to bring out the scientist in every student.  Its tentacles reached far and wide to Europe and developing countries, where suitability to local conditions  invariably posed a problem, as both the teachers, parents and students were ill-prepared to handle the the high-level abstraction,  perhaps appreciated (and understood) only by those headed for university physics or math.

          Discontent with the  impracticability of the New Math to most students, a third seismic shift  began in the late 1970s, spilling into the next 2 decades.  Back-to-basics, mandated national standards, and  teacher and school accountability were the new buzz words.  NCTM  played a stellar role in the US when it developed its own standards program after innumerable consultations with education stakeholders:  math teachers, parents, administrators, education specialists.  NCTM's huge impact is such that it served as a model to develop curricular standards for other subject areas, as well as serving a focal reference for many of the states in the US, in Canada, as well as developing countries.  NCTM is heavily leaned on the progressive end where sense-making, original thinking, conceptual understanding, exploratory process and meaningful problem solving are the emphasized more than teacher-constructed algorithm, fluency, end-answers.

       As good as this sounds, discontent will always be there, which is expected as we adapt to changing national and global scenarios.  Such was the case when the TIMSS results showed dismal performance of  the US and the traditional superpowers, compared to the top performance of recent world players and emerging economies of  Singapore, Taiwan and South Korea.  Some western economies are closely monitoring  Asian classrooms, and how they can possibly import some lessons in their quest for a better math education.

How to Make a Hole in Postcard-sized Paper Big Enough to Pass Through

 

What	Time	Materials
Greet students. Today we'll explore one of the most pefect shapes in nature. Can you guess what	5 s
this may be? (Circle)
At the end of the lesson, students will be able to:
1. have a deeper appreciation of the area of a circles
2. problem solve creatively
3. think outside the box
1. to motivate students as to be 100% engaged
2. to elicit maximum participation involving the 3 senses (visual, auditory, touch)
1. Can everyone construct a circle with their 2 fingers? 3 fingers? 4 fingers? 5 fingers? 10 fingers?
2. Was making a good circle easy? Compared to say makng/drawing a sqaure or a rectangle? Why?
3. Compare the circles formed by someone with shorter fingers with a person with longer fingers.
What do you call the area inside the circle/hole?
1. Let's explore the biggest circle/hole we can make out of a piece of paper. Who has some idea of	7 m	whiteboard,
this is done? Show in the whiteboard.		markers
2. Discuss: areas of a unit circle vs . 2x2 circle vs. 3x3 circle.
3. Draw conclusions (The bigger the paper, the bigger the circle that can be cut in it.		10 postcard-size
4. Show a piece of postcard-size paper. What is the biggest circle/hole you can make out of it.		cardboard
Is the hole large enough to put your fist through? Your head through? For the whole length of you		with pre-drawn
pass through?		line
5. What if I tell you a secret that it is possible to cut a hole through this cardboard big enough for you		markings
to pass through? What woud the pattern of the cutting be like? (zigzag/irregular) Why?
6. Give each student a postcard with pre-drawn line markings. Ask them to describe the pattern		scissors
of dotted lines. Ask them to visualize the resulting hole this pattern will produce.
7. Give general instructions on how to cut the paper. Guide each student as they do the actual cuts.
8. End result: a huge hole big enough for a person to pass through.
1. How did you use the concept of area to cut a super-sized hole out of a small paper?	1 m
1. Summarize activity.	2 m

Teacher/Student Interviews

TEACHING AND LEARNING MATHEMATICS

It is well known that good teachers love the subject they are teaching.  On the other hand, if you as the teacher feel negative towards mathematics, it may show up when you are teaching your students and can affect them similarly.  Little children usually like numbers and math - yet many kids in schools develop 'math anxiety or phobia' or end up disliking math.  A major factor in that is the way math is taught and the way the teachers feel about math.

We have been to a Senior Secondary School, and asked several questions to both math’s teachers and students. The results were not that surprising. After interviewing several students, we found that most of them do not like mathematics and find it boring. By analyzing different students, we realized that they want to be taught by instrumental way. However, they want to understand purpose of math they are studying. They could not relate the purpose of learning math other than measuring, estimating the bills. Even when they are asked that what they want to change in curriculum, their answer was that want math which can be used in daily life. Therefore, the students might get more motivated if she/he knows where all maths is needed.  So many times kids question the needfulness of things they study.  Emphasizing and pointing out the everyday applications of math may help them. Even when they are asked that what they want to change in curriculum, their answer was that want math which can be used in daily life.

Also, students prefer hands on learning rather than lecture method. They like teachers who involve activities in their lessons. One student said that she wants math more hands on which means that they want more activity oriented curriculum. Thus, by including different activities in class we can motivate students.

We had a chance to interview and observe secondary mathematics teacher in a class. Students were taking interest and paying attention to what they were taught.  We talked about her teaching strategies. She told us that  one very important factor in motivating students to study math is that you yourself, as the teacher, stay positive about math - if possible, enthusiastic! .

Secondly, we need to get the student involved!  One of the reasons for math anxiety is the way math is often taught as "There is only ONE way to do this, and you need to do learn it and do it right."  Math is presented as 'given from above'.  Students can be much more motivated if they are asked open questions, involved in the development of concepts, given very open-ended exercises.  Granted, this kind of teaching style may require a lot of planning from the teacher, probably a good understanding in math, and good materials.

Thirdly, the teacher should not put a wrong answer down.  Instead, say, "Please can you explain how you came up with that?"  In a classroom, a teacher can ask, "Did someone else get the same result as you? OK. Did somebody get a different result? OK, we have two (or three) different answers here. Let's figure them out."  Wrong answers are valuable.  You get insight into student's thinking and where he went wrong, and what needs a rethought.  Students and kids need to be treated as humans and not feel put down or stupid for their answers.

Last but not least,take the emphasis off from tests.  Tests are a part of school but they don't need to be the ultimate goal.  She told us that, she returns the quizzes back to students and ask them to do the corrections. The goal is to learn math so the child can use it in her life. 
IT’S THE ATTITUDE NOT THE APTITUDE THAT DECIDE THE ALTITUDE OF YOUR SUCCESS.

Non-math microteaching

Strength's of the group's presentation:  1.  Variety on interesting topics  2.  Engaging personalities of presenters   3.   Uncomplicated activity suitable for a 10-minute time frame   4.  a plan B in case there's spare time
My presentation:  Wasn't too happy as I messed up with my main material.  Also, poor time management.
Lessons learned:  Be sure to pre-try your planned  activity at least twice.  Choose a suitable topic within the given time frame.

Dave Hewitt: Model Math Teacher #2

Dave Hewitt's totally unconventional approach to teaching blew my mind away, something akin to a 20-minute medium-sized twister.  He defied a lot of hallowed (read traditional) conventions of teaching.  First, he ignored the primary function of the whiteboard for the better part of the lesson and  used it instead as a rhythmic background sound for his imaginary number line.  One can doze off copying notes but not with a regular, very audible tapping in the board.  Besides, Dave hardly takes his eyes off his students, as if instantaneously processing & learning from each moment. Second, he circled the room with his stick, tapping back & forth, to teach skills with increasing complexity.  One can never be sure what's next; anticipation, hence engagement is high.  Third, the lessons progressed seamlessly to the physical placement of integers in the number line, to addition/subtraction of integer(s), to equivalent expressions, to the sneaky way of introducing variables, to solving equations using properties of equalities & inverse operations, & order of operations. The beauty is that the increasing complexity of the lesson appeared to be the most natural way an average person can think for himself; there were no explicit definitions, rules, or algorithms given as a reference starting point.  It takes a teacher great courage, insight & dedication to want to do untested things.  But we owe it to our students and ourselves to be in a perpetual state of creative dissatisfaction to continually find better ways to teach and learn.  Bill Gates recently said that if you want your child to be educated well, it is not as important to send him to a good school than to a great teacher.

Skemp article

Reaction to article:  Relational Understanding and Instrumental Understanding, R. Skemp

The concept of faux amis in math teaching and learning is a very valid and powerful one.  Valid because of the high preponderance of instrumental learning even in today's math  education (despite the research) ; powerful because a shift to relational learning requires nothing short of a 360 turn of mental paradigm.

Universities and business have decried the lack of creativity, logical reasoning, problem solving mindset (e.g. correctly defining the problem, conceptual mapping of a problem into its interrelated parts, looking at a problem from different perspectives, persistence) of our graduates.  A major cause would be the instrumental approach of most teachers, a view necessarily adopted by students, and implicitly encouraged by the parents (who enrol children in tutorial classes & demand top marks), by the Education Ministry and some universities, that measure mathematical knowledge with mandatory provincial & entrance exams, the bulk of the questions testing instrumental “understanding.”

In my 12 months tutoring K-12 Math at Sylvan Learning, I would many times encounter tutees who are only interested in the “rules,” apply them automatically, with some success to routine problems, and very little success to non-routine ones.  When discussing their errors, I would launch into the rationale behind the rules, but most of them tune out, as if they don't want their minds burdened by more math.  Hence students are happy to be reminded that “you flip inequality signs when multiplying/dividing by negative numbers” or that “you can cancel same factors in the top & bottom numbers of fractions, but not same addends.”  Not one of them asked why this is so. 

Part of the problem seems to be the approach of some of their previous teachers, who introduce the rules early on the topic, and follow these with drill & kill exercises, for skill mastery.  Such was the case when exponents were “discussed” in my Grade 10 son's class.  He knew that he had to add exponents when multiplying similar bases because of the rule, but couldn't explain why. Moreover, he was at a lost why you now multiply exponents when you raise a power to a power.  The interesting part was that when I tried to explain the raison-de-etat of the rules, he tuned out because “that's not the way the teacher explained it.”  In other words, he accepted the teacher's approach with utmost faith, that no other approach or explanation is possible. (or could it be because I'm just a parent & not an official teacher?)

The same thing happened in his elementary math when he asked me if he correctly calculated the area of an irregular figure where an isosceles trapezoid was nested.  For starters, I asked him why the formula of the trapezoid was such, he replied that he recalled his teacher dissecting the trapezoid & moving bits around, but he did not copy the drawing since  “it is only the formula that matters.”  And you guessed it right, he was not receptive to any explanation on my part.

So even in this instance when the teacher was going for relational understanding, such effort was undermined.  I can offer 2 possible explanations:  first, the follow-up assignments and the all-important tests were just plugging into the formula.  Second, the teacher might in the majority of his past lessons explained instrumentally (with matching tests).

Memorable Math Teachers

High school and college went by without a single Math teacher impacting hugely on my consciousness.  All teachers were basically the same, the majority droned endlessly with their expert renditions of the day's topic.  You either get it or you don't, if you're in the latter category, you're pretty much on your own.  Then came my student practicum mentor.  She's a single mom who has taught in the US with her young child in tow.  As a single parent, she needed to reevaluate her priorities and options, and question and look at things from a different perspective.  And that's how she approached the teaching of Math.
Given a mandated curriculum to implement, she looked for creative ways to present a flow of lessons that will maximize student engagement & discuss a richer depth of the topic.