Tag Archives: Jo Boaler

Low Floor and High Ceiling Math

Whether you like to solve problems yourself, or are looking for some tidbits for your children or students, I hope this post is informative.

I’ve been reading Jo Boaler‘s brilliant new book, Mathematical Mindsets.  While there’s tons of great information and research there, I’ve been thinking lots lately about her charge to develop more “low floor, high ceiling” tasks into math lessons–problems that are “challenging, but accessible” to a much broader spectrum of students than typical exercises.  In particular, Boaler encourages teachers to use problems that are easily understood, relatively simple to begin, and yet hold deep potential for advanced exploration.  Boaler notes that these problems tend to be very difficult to find.

Here I offer an adaptation of an Ask Marilyn post toward this goal.  While the problem was initially posed in terms of singles at a party; I rephrased it for younger students.  Solving it helped me see variations that I hope address Boaler’s low floor, high ceiling call.

THE PROBLEM :

Paraphrasing the original:

Say 100 students stop by the lunchroom for a snack.  Of these, 90 like apples, 80 like pears, 70 like bananas, and 60 like peaches.  At the very least, how many students like all four fruits?

MY FIRST SOLUTION:

The Ask Marilyn post offered only the answer–zero–but not a solution.  To prove that, I made picture.  Since the question was the least number of commonly liked fruits, I needed to spread out the likes as much as possible.  Ninety liked apples, so when I added the pears, I made sure to include the 10 non-apples among the 80 who did like pears giving

fruit1

That made 30 (at the bottom) who liked only one of apples or pears, so when I added the 70 bananas, I first added them to those 30, leaving

fruit2

That made 40 who liked all three, so the 60 peaches could match up to the other 60 who liked only two of the first three, confirming vos Savant’s claim that it was possible in this setup to have no one liking all four.

fruit3

THINKING OUTWARD:

FIRST:  As a minor extension, one of my students last year would have said the problem could be “complexified” slightly by changing the numbers to percentages.  (I loved my conversations with that student about complexifying vs. simplifying problems to find deeper connections and extensions.)  With enough number sense, students should eventually be able to work with absolute numbers and relative percentages with equal ease.  Mathematically, it doesn’t change anything about the problem.

SECOND:  The problem doesn’t have to be about a single minimum number of students to like all four fruits.  While there is a unique minimum, there are many other non-optimal arrangements.  I wonder how students with developing problem solving skills would approach this.

THIRD:  In my initial attempts, I had used many different variations on my tabular solution above.  Only in the writing of this post did I actually use the above arrangement, and that happened only because I was trying to come up with a visually simple representation.  In doing so, I realized that the critical information here was not what was told, but what was not said.  Where 90, 80, 70, & 60 liked the given fruits, that meant a respective 10, 20, 30, & 40 did not.  And those added up to 100, so I knew that any variation of “not-likes” that also added to 100 could be distributed so that the minimum number who liked all four would also be zero.  So there is an infinite number (if I use percentages) of variations of this problem that have the same answer.  I also realized that any combination of 2 or more fruits whose “not likes” added to 100 could produce the same results.  My ceiling just rose!

FOURTH:  To make the problem more accessible, I could rephrase this in terms of setting out fruit and exploring many different possible arrangements.  i could also encourage learners to support their developing problem solving by translating the problem into pictures.

I’m ready to pose a new variation.  I’d love to hear your thoughts, insights, and variations for raising the ceiling in this problem.

LOW FLOOR, HIGH CEILING VARIATION?

Say 100 students stop by the lunchroom for a snack.  Of these, 90 like apples, 80 like pears, 70 like bananas, and 60 like peaches.  The lunchroom staff knows these numbers, but doesn’t know how much of each fruit to put out.  But putting out too much fruit would be wasteful.

  • What advice can you give them?  Show how you know your solutions are correct.
  • Draw some pictures of the possible numbers of students who like the different fruits.
  • Is there more than one possible answer?
  • It is possible that some students might not like any fruits offered.  How many students might this describe?
  • Some students might like all four fruits.  For how many students might this be true?  How many answers are there to this?

For students who manage all of these, you can challenge them to

  • How can you change the initial numbers in this problem without changing most of the answers?
  • Can you create the same scenarios with more or fewer types of fruit?
  • If the numbers are too big for very young students, you could drop all of the initial numbers by a factor of 10.  How many will see this scaling down simplification (or its scaling up complexification)?

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.

Boaler

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.

Robinsons1

Robinsons2

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?