# Tag Archives: pedagogy

## Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.

I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.

## Which came first: Math Ability or Computational Speed ?

I’ve claimed many times in conversations over the last two weeks that I believe many parents and educators misconstrue the relationship and causality direction between being skilled/fluent at mathematics and being fast at computations.  Read that latter as student accomplishment defined by skill on speed testing as done in many, many schools.  Here is a post from Stanford’s Jo Boaler on math anxiety created by timed testing.

Here’s my thinking:  When we watch someone perform at a very high level in anything, that person appears to perform complex tasks quickly and effortlessly, and indeed, they do.  But . . . they are fast because they are good, and NOT the other way around.  When you learn anything very well and deeply, you get faster.  But if you practice faster and faster, you don’t necessarily get better.

I fear too many educators and parents are confusing what comes first.  From my point of view, understanding must come first.  Playing with ideas in different contexts eventually leads to recognizing that the work one does in earlier, familiar situations eventually informs your understanding in current, less familiar settings.  And you process more quickly in the new environment precisely because you already understood more deeply.

I think many errantly believe they can help young people become more talented in mathematics by requiring them to emulate the actions of those already accomplished in math via rapid problem solving.  I worry this emphasis is placed in exactly the wrong place.  Asking learners to perform quickly tasks which they don’t fully understand instills unnecessary anxiety (according to Boaler’s research) and confuses the deep thinking, pattern recognition, and problem solving of mathematics with rapid arithmetic and symbolic manipulation.

Jo Boaler’s research above clearly addresses the resulting math anxiety in a broad spectrum of students—both weak and accomplished.  My point is that timed testing–especially timed skill testing–at best confuses young students about the nature of mathematics, and at worst convinces them that they can’t be good at it.  No matter what, it scares them.   And what good does that accomplish?

## Gender and Expectations Lessons from Research

A couple reports from NPR yesterday have me thinking about some of the articles I’ve compiling in my Diigo library about what I’ve learned about gender and expectations differences in parenting and teaching.  I don’t have anything particular to tie together here, but I thought these four resources were more than I could comfortably tie together in a coherent Tweet thread, so I thought I’d gather them into an impromptu ‘blog post.

Girls May Get More ‘Teaching Time’ From Parents Than Boys Do via @NPR.  Excerpts:
… ” ‘How often do you read with your child?’ or ‘Do you teach them the alphabet or numbers?’ … Systematically parents spent more time doing these activities with girls.”
… “Since parents say they spend the same amount of time overall with boys and girls, Baker’s analysis suggests that if parents are spending more time with girls on cognitive activities, they must be spending more time with boys on other kinds of activities.”
… “The costs of investing in cognitive activities is different when it comes to boys and girls. As an economist, he isn’t referring to cost in the sense of cash; he means cost in the sense of effort.”

Gender Gap Disappears in School Math Competitions via sciencedaily.com .  Excerpt:
… “Most school math contests are one-shot events where girls underperform relative to their male classmates. But a new study by a Brigham Young University economist presents a different picture.  Twenty-four local elementary schools changed the format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.”
… “It’s really encouraging that seemingly large gaps disappear just by keeping [girls] in the game longer.”

A broader look at school expectations leading to enhanced math performance:  What Distinguishes a Superschool From the Rest via ideas.time.com .
… “The difference seems to lie in whether a school focuses on basic competence or encourages exceptional achievement. While almost all the schools saw it as their responsibility to cover the math knowledge necessary to do well on the SATs, the authors noted that “there is much less uniformity in whether schools encourage gifted students to develop more advanced problem solving skills and reach the higher level of mastery of high school mathematics.”
… “The fact that the highest achieving girls in the U.S. are concentrated in a very small set of schools, the authors write, indicates ‘that almost all girls with the ability to reach high math achievement levels are not doing so.’ ”

Girls, Boys And Toys: Rethinking Stereotypes In What Kids Play With via @NPR.  Excerpts:
… Some toy companies are re-thinking gender-specific marketing and branding.
… “I think what they were worried about was causing gender identification needlessly — to turn off passive learning, passive expression down the road, even passive economic opportunity for girls or boys if they felt they couldn’t do something because of societal norms,”
… “It’ll be interesting to see how this changes the attitudes of parents and of kids over time or whether or not it does. There may be some hard-wired differences,”

## Binomial Probability and CAS

I posted previously about a year ago an idea for using CAS in a statistics course with probability.  I’ve finally had an opportunity to use it with students in my senior one-semester statistics course over the last few weeks, so I thought I’d share some refinements.  To demonstrate the mathematics, I’ll use the following problem situation.

Assume in a given country that women represent 40% of the total work force.  A company in that country has 10 employees, only 2 of which are women.
1) What is the probability that by pure chance a 10-employee company in that country might employ exactly 2 women?
2) What is the probability that by pure chance a 10-employee company in that country might employ 2 or fewer women?

Over a decade ago, I used binomial probability situations like this as an application of polynomial expansions, tapping Pascal’s Triangle and combinatorics to find the number of ways a group of exactly 2 women can appear in a total group size of 10.  Historically, I encouraged students to approach this problem by defining m=men and w=women and expand $(m+w)^{10}$ where the exponent was the number of employees, or more generally, the number of trials.  Because question 1 asks about the probability of exactly 2 women, I was interested in the specific term in the binomial expansion that contained $w^2$.  Whether you use Pascal’s Triangle or combinations, that term is $45w^2m^8$.  Substituting in given percentages of women and men in the workforce, $P(w)=0.4$ and $P(m)=0.6$, answers the first question.  I used a TI-nSpire to determine that there is a 12.1% chance of this.

That was 10-20 years ago and I hadn’t taught a statistics course in a very long time.  I suspect most statistics classes using TI-nSpires (CAS or non-CAS) today use the binompdf command to get this probability.

The slight differences in the input parameters determine whether you get the probability of the single event or the probabilities for all of the events in the entire sample space.  The challenge for the latter is remembering that the order of the probabilities starts at 0 occurrences of the event whose probability is defined by the second parameter.  Counting over carefully from the correct end of the sequence gives the desired probability.

With my exploration of CAS in the classroom over the past decade, I saw this problem very differently when I posted last year.  The binompdf command works well, but you need to remember what the outputs mean.  The earlier algebra does this, but it is clearly more cumbersome.  Together, all of this screams (IMO) for a CAS.  A CAS could enable me to see the number of ways each event in the sample space could occur.  The TI-nSpire CAS‘s output using an expand command follows.

The cool part is that all 11 terms in this expansion appear simultaneously.  It would be nice if I could see all of the terms at once, but a little scrolling leads to the highlighted term which could then be evaluated using a substitute command.

The insight from my previous post was that when expanding binomials, any coefficients of the individual terms “received” the same exponents as the individual variables in the expansion.  With that in mind, I repeated the expansion.

The resulting polynomial now shows all the possible combinations of men and women, but now each coefficient is the probability of its corresponding event.  In other words, in a single command this approach defines the entire probability distribution!  The highlighted portion above shows the answer to question 1 in a single step.

Last week one of my students reminded me that TI-nSpire CAS variables need not be restricted to a single character.  Some didn’t like the extra typing, but others really liked the fully descriptive output.

To answer question 2, TI-nSpire users could add up the individual binompdf outputs -OR- use a binomcdf command.

This gets the answer quickly, but suffers somewhat from the lack of descriptives noted earlier.  Some of my students this year preferred to copy the binomial expansion terms from the CAS expand command results above, delete the variable terms, and sum the results.  Then one suggested a cool way around the somewhat cumbersome algebra would be to substitute 1s for both variables.

CONCLUSION:  I’ve loved the way my students have developed a very organic understanding of binomial probabilities over this last unit.  They are using technology as a scaffold to support cumbersome, repetitive computations and have enhanced in a few directions my initial presentations of optional ways to incorporate CAS.  This is technology serving its appropriate role as a supporter of student learning.

OTHER CAS:  I focused on the TI-nSpire CAS for the examples above because that is the technology is my students have.  Obviously any CAS system would do.  For a free, Web-based CAS system, I always investigate what Wolfram Alpha has to offer.  Surprisingly, it didn’t deal well with the expanded variable names in $(0.4women+0.6men)^{10}$.  Perhaps I could have used a syntax variation, but what to do wasn’t intuitive, so I simplified the variables here to get

Huge Pro:  The entire probability distribution with its descriptors is shown.
Very minor Con:  Variables aren’t as fully readable as with the fully expanded variables on the nSpire CAS.

## What would you teach?

I’d love some insights for the statistics content of a non-AP senior math course.

BACKGROUND: I’ve been asked to step into a combination terminal course for seniors covering an introduction to statistics in the fall semester and an introduction to calculus in the spring.  The course has no prerequisite and most of its students have seen precious little high school statistics or probability beyond model regressions to bivariate data in 10th grade.  The class is populated almost entirely by students who are very smart, but who’ve never experienced an honors math course.  A large portion have been frustrated and unmotivated by their previous math courses; for some, this is the last math course they ever take.

The very broad purpose is to introduce students to both branches to learn the fundamentals of what each does.  The department’s pitch for the course when it was created last year acknowledged that we could not know whether our students would be required to take calculus or statistics once they got to college, so this was the “best” way we could prepare students for college mathematics.  It meets 4 days/week for 55 minutes/session.

REFINED QUESTION:   Imagine you were to teach statistics for exactly one semester with no external limitations beyond what has already been described.  What would you make the key focal points for your class?  Why?

## The Problem with the Test

Following is a snip from a 12.19.2011 post by James K. to the AP Calculus EDG .  It was a response to a teacher’s frustration with her district increasing the number of students in her AP Calculus course and then evaluating her performance in part on the scores her students receive AP exam.  Sorry for the length of the block quote, but it’s worth absorbing.

I have some separate thoughts about tests and teacher evaluation, which I’m hoping to carve out time for sometime soon, but I’m eager to develop the conversation around the following comments:

[citing another post] … I do get students with weaker and weaker math backgrounds.  How weak? I have student who can’t evaluate sin(pi/3), and a student [who] can’t understand it’s the same to multiply a number by 1/6 and to divide it by 6. How did they pass precalculus? …. It feels nice to be able to say “I’m always willing to teach those willing to learn”, but how do you teach a class with huge gap in students’ capability? Teach fast, and you get lots of complaints. Teach slower, and the more capable students suffer. They (the more capable) are the ones that pay the price when we say “I’m always willing to …” [end internal citation]

What’s at the heart of this discussion is differentiating instruction: that is, developing and implementing activities (which can include lecture, as just one type of activity) which provide rich, effective learning for a wide variety of students.  This requires acknowledging not only that students come to the class with diverse background knowledge and conceptual frames, but also that they are likely to leave the class with different knowledges and conceptual frames– in short, serving [intellectually] diverse students … means valuing significant growth at least as much as valuing specific pre-determined objectives.

This is, I believe, the greatest and most tragic failure of mathematics instruction over the past 4000 years: the teaching of mathematics almost invariably has been designed and implemented around an assumption that the cognitive development of mathematical ability is linear in nature, with the overwhelming result that math courses are substantially about distinguishing successful students (those who are able to reproduce a specific set of mathematical procedures and algorithms in novel contexts) from unsuccessful students (who are unable to do so) … as opposed to being about the development of better mathematical thinking in all of the students in the course.

That’s gotten a bit better, just in the past 100 years or so, but the culture is still overwhelmingly focused on determining who can “do the math” (and, implicitly, who cannot).  I recognize that, particularly within the context of an AP course (where the desired outcomes of the course are richly and rigorously defined by an external examination), it’s hard to embrace the notion that it’s okay for students to come away from a lesson, a unit, and the entire course as a whole with different degrees of knowledge and understanding, and even with different ways of *approaching* their own understanding.  That said, I think it’s the work that we ought to be doing:  Teaching is about fostering growth.  It’s certainly easier to do with a roughly uniform group– but I don’t let my students get away with ignoring the hard stuff, and I don’t let myself get away with it, either.  That means we ought to be teaching at both (really, all) levels, simultaneously: give the weaker students what THEY need, while also giving the more capable students what THEY’RE ready for, all at once.  Not easy, as I say, and especially hard to do through lecture (which is why that’s just one of the types of activity we should be using…)

I certainly haven’t achieved all I want to do with respect to differentiating instruction in my classes.  I’ve also been part of numerous discussions like the one James advances above.  His argument for reaching all students where thy are with what they need is, in my opinion, the great calling of all teachers.

What struck me as unique about James’ post was his focus on interpreting the outcome of an AP Course.  I’ve never clearly thought about it in this way, but the standardization of a single international test for all students with only five possible outcomes (scores ranging 1 to 5) does drive one to believe that all students (if they would just try) will exit the course with the same curricular and comprehension results.  I’ve worked for years (with varying success) to meet my students in their areas of need and have tried to promote their unique perspectives.  I hope all of my students (no matter what their mathematical background) say my classes push them to think in new ways and encourage them to apply what they have learned in ways unique to their own perspectives.  What I hadn’t fully considered was the pervasive way external tests can leverage teachers away from that ideal.

So, I recommit myself to

the notion that it’s okay for students to come away from a lesson, a unit, and the entire course as a whole with different degrees of knowledge and understanding, and even with different ways of *approaching* their own understanding,

especially when teaching courses with external final assessments.

## Transforming inverse trig graphs

It all started when I tried to get an interesting variation on graphs of inverse trigonometric functions.  Tiring of constant scale changes and translations of inverse trig graphs, I tried $i(x)=x*tan^{-1}x$ , thinking that this product of odd functions leading to an even function would be a nice, but minor, extension for my students.

I reasoned that because the magnitude of arctangent approached $\frac{\pi}{2}$ as $x\rightarrow\infty$, the graph of $i(x)=x*tan^{-1}x$ must approach $y=\frac{\pi}{2}|x|$ .  As shown and to my surprise, $y=i(x)$ seemed to parallel the anticipated absolute value function instead of approaching it.  Hmmmm…..

If this is actually true, then the gap between $i(x)=x*tan^{-1}x$ and $y=\frac{\pi}{2}|x|$ must be constant.  I suspected that this was probably beyond the abilities of my precalculus students, but with my CAS in hand, I (and they) could compute that limit anyway.

Now that was just too pretty to leave alone.  Because the values of x are positive for the limit, this becomes $\displaystyle y=\frac{\pi}{2}|x|-x*tan^{-1}x=\frac{\pi}{2}x-x*tan^{-1}x=x*(\frac{\pi}{2}-tan^{-1}x)$ .

So, four things my students should see here (with guidance, if necessary) are

1. $i(x)=x*tan^{-1}x$ actually approaches $y=\frac{\pi}{2}|x|-1$,
2. the limit can be expressed as a product,
3. each of the terms in the product describes what is happening to the individual terms of the factors of $i(x)$ as x approaches infinity, and
4. (disturbingly) this limit seems to approach $\infty*0$.  A less-obvious recognition  is that as $x\rightarrow\infty$, $\frac{\pi}{2}-tan^{-1}x$  must behave exactly like $\frac{1}{x}$ because its product with x becomes 1,

But what do I do with this for my precalculus students?

NOTE:  As a calculus teacher, I immediately recognized the $\infty*0$ product as a precursor to L’Hopital’s rule.

$\displaystyle\lim_{x\to\infty} [x*(\frac{\pi}{2}-tan^{-1}x)]\rightarrow\infty*0$
$=\displaystyle\lim_{x\to\infty}\frac{\frac{\pi}{2}-tan^{-1}x}{\frac{1}{x}}\rightarrow\frac{0}{0}$
and this form permits L’Hopital’s rule
$=\displaystyle\lim_{x\to\infty}\frac{\frac{-1}{\displaystyle 1+x^2}}{\frac{-1}{\displaystyle x^2}}$
$\displaystyle=\lim_{x\to\infty}\displaystyle\frac{x^2}{1+x^2}=1$

OK, that proves what the graph suggests and the CAS computes.  Rather than leaving students frustrated with a point in a problem that they couldn’t get past (determining the gap between the suspected and actual limits), the CAS kept the problem within reach.  Satisfying enough for some, I suspect, but I’d love suggestions on how to make this particular limit more attainable for students without invoking calculus.  Ideas, anyone?

## CAS for All

Every student should have access to a CAS (Computer Algebra System) in a handheld and/or computer-based format at least as early as he or she begins learning algebraic concepts.

Used properly, a CAS creates a dynamic laboratory environment for a student in which he or she can explore algebraic relationships, receive  instantaneous confirmation of the validity of algebraic manipulations, and scaffolding for deeper exploration and understanding of mathematics.  In short, a CAS enables a student to have a mathematical solving expert available at all times in all places.  Most importantly, students get the opportunity to explore mathematics without needing

Of course, to use a CAS, one needs to learn how to ask questions and how to interpret the solutions.  A CAS will always provide an answer to the question asked.  Users must know precisely what is being asked so that they can interpret their results.