My Math Education Blog

"There is no one way"

Monday, February 13, 2017


Many students have weak arithmetic skills. Many teachers blame this on calculator use, but it is just as likely that the real reason lies elsewhere. For one thing, the teaching of arithmetic traditionally does not involve developing any understanding, so the learning is shallow and fragile. For another, students correctly feel that mindless arithmetic is no longer a useful skill in the age of technology, so it may not be so much calculator use, but the very existence of of the calculator which saps motivation in this arena. Finally, there was a time when many high school and college teachers didn’t need to interact with students whose arithmetic skills were weak, because those students were prevented from taking college-preparatory math classes. This is no longer true, and the population of college-intending students has grown enormously, so those teachers erroneously conclude that arithmetic skills are getting worse. In any case, in my view, there is no reason to ban calculator use in the classroom. Such a rule will be perceived as punitive and only build resentment and a negative disposition. On the other hand, students have no problem at all accepting a ban on calculators during specific activities, (such as a mental math session, or a quiz on famous trig values) because in that context, the ban is readily explained and eminently sensible. 


If you wanted to know the result of multiplying 5463 by 78912, it is very likely that you would reach for a calculator. No one in the world outside of schools would do this problem with pencil and paper. Children know this. They know that adults at work do arithmetic by machine, whether in a fast food joint, or a bank, or a lab, or an engineering firm – anywhere at all. It is obvious to students that doing multidigit arithmetic by hand is not a useful skill. No one will hire you to do long division. 

Still, being able to predict that the result of that multiplication is not too far from 400,000,000 is a worthwhile skill, if only to confirm that the answer given by a calculator is in the right range. I got this result by multiplying 5000 by 80,000. I could gain a little more accuracy by following this up with an estimate for 55 × 79. Well, 55 × 80 is (5 × 80) + (50 × 80), or 4400. Subtract 55, and get 4345. So the result of the original calculation should be close to 434,500,000. 

How did I do? The actual answer is 431,096,256. In millions, my error was a number less than 4, divided by a number greater than 400. In other words, I was within 1%. Not too bad! (My calculator says the error was .7895555%.) 

Few students would be able to work through this like I did. That is not surprising: this sort of computation is not taught much. If students spent less time with paper and pencil multi-digit arithmetic, that would free up some time to work on mental calculation and estimation. Working on this depends on, and helps develop number sense. For my first estimate, I had to understand rounding, and multiplication by powers of ten. In the second phase I used the distributive law twice. Paper-pencil computation, on the other hand, can be done with little or even no understanding. And almost no students gain deeper understanding by doing it. 


But, you say, by practicing multi-digit multiplication on paper, they can gain accuracy and speed! Yes, that is true of many students. But even the best will not be as accurate or as fast as the free app on their phone. And in any case, what good would it do? Speed and accuracy in computation can no longer be a priority in math education. We should spend what limited time we have with students developing their understanding, not wasting it on useless skills. 

Mental calculation and estimation need a place of honor throughout the K-12 curriculum As far as I’m concerned, that is the main consequence of the availability of calculators. This can take place in number talks appropriate to what is being taught at each level. Estimates should precede every calculation. Possible strategies should be discussed before and after the introduction of standard or non-standard algorithms. 

And this need not stop after middle school, quite the opposite: number sense and operation sense continue to be a priority all the way to graduation and beyond. What is your estimate for the square root of 18? why? A mental math component can be added to topics like algebra: solve 3x + 1 = 13 without paper, pencil, or technology. Or trigonometry: you should know or be able to quickly retrieve cos(60). What would you guess for cos(55)? cos(65)? 

One thing that the traditional paper-pencil approach and the mental math approach have in common is that they both require knowing basic addition and multiplication facts. So yes, students should learn their addition and multiplication tables. Of course, those facts are best learned in the context of understanding. 

But what about students for whom remembering 6 × 7 is an unreachable goal? No matter what we try, they can’t seem to hang on to those facts. Should we shut the door in their face, and say that if you don’t know your multiplication table you will not be allowed to go on with secondary school math? That is the way it used to be. I don’t think such a policy makes sense any more. If a student needs to use the calculator ”as a crutch”, I let them. I wouldn’t deny a crutch to someone with a broken leg. 

None of this is to deny that there is some interesting math to learn when working, within reason, on three-digit addition, or two-digit multiplication. But that is my point: learning some interesting math should be the goal. Speed and accuracy will ensue for some students, but it should not be prioritized.

Thursday, February 9, 2017

Geometry Boot Camp!

I will offer two workshops this summer (2017), at the Head-Royce School in Oakland, CA.
Sign up for either or both!
June 26-27: Hands-On Geometry (grades 6-10)
June 28-30: Transformational Geometry (grades 8-11)
If the times or locations don't work for you, I can offer a workshop for your school or district. Contact me directly.
I recommend attending with a colleague, as it makes it easier
to implement these ideas when you get back to school in the fall.
Support materials are on the workshop participants' Web site.

Hands-On Geometry

with Henri Picciotto Monday-Tuesday, June 26-27
9:00 a.m. to 3:30 p.m.
at the Head-Royce School in Oakland, CA


In this two-day workshop for teachers in grades 6-10, I will present kinesthetic and manipulative activities. This hands-on curriculum is intended to complement related work in paper-pencil environments: it serves to preview, review or extend key concepts in geometry. The activities can be used to enrich and enliven the high school geometry course, or to lay the groundwork for it.
  • Topics include angles, triangles, quadrilaterals, area, the Pythagorean theorem, congruence, similarity, "soccer angles", and tiling.
  • Tools include manipulatives (such as pattern blocks and geoboards) and puzzles (such as tangrams and pentominoes.)
  • Technology will be used to illustrate concepts.
These lessons were developed in somewhat heterogeneous classes, and reach a wide range of students. They provide support for the less visual by complementing the drawing and studying of figures, and enrichment for the more talented by offering deep and challenging problems.

This workshop does not overlap with Transformational Geometry.

Transformational Geometry

with Henri Picciotto Wednesday-Friday, June 28-30
9:00 a.m. to 3:30 p.m.
at the Head-Royce School in Oakland, CA


The Common Core State Standards call for a complete rethinking of geometry in grades 8-11. Instead of basing everything on congruence and similarity postulates, as is traditional, the idea is to build on a foundation of geometric transformations: translation, rotation, reflection, and dilation.

In order to teach this effectively, it is important to have a solid understanding of the underlying math, as well as ideas for rich activities for students. I have been teaching transformational geometry for twenty years, and have a lot to share. This three-day workshop will cover:
  • The implications for the teaching of proof.
  • Composition of transformations -- the four isometries and their fundamental properties.
  • Symmetry in depth -- around a point, along a strip, in the plane. Connections to art and design.
  • Computing geometric transformations with the help of complex numbers at first, then matrices -- this is the mathematics that underlies all computer graphics.
  • Transforming graphs: all parabolas are similar
  • Intelligent use of technology to support all this, including a highly motivating unit on geometric construction.
This workshop does not overlap with Hands-On Geometry


The Details

Check this site in late February for Continuing Education Units info.
More info about me: Henri Picciotto
Registration and logistics: Head-Royce School 
Questions? Send me e-mail
Scholarships: an anonymous donor will pay 80% of the tuition for the first seven public school teachers to register.
If the times or locations don't work for you, I can offer a workshop for your school or district.  
Contact me directly.


Wednesday, February 1, 2017

Comparing two approaches

Much can be said in defense of practice exercises, but when all is said and done, very few students develop deep understanding from routine practice. For example, compare these two approaches to the area of a trapezoid.

Approach 1

The teacher says: ”The area of a trapezoid is given by the formula h(b1+b2)/2, where h is the height, b1 and b2 are the bases. Here is a worksheet where you can practice this.” 

The worksheet includes 20 examples, each with different numbers for the bases and the height. The students practice in silence. Many students like this, because they know exactly what to do. Other students don’t like it, because they find it boring. All of them know they will soon have to calculate some trapezoid areas on a quiz, and they hope that this practice will help them remember the formula. Some will make an effort to memorize the formula in preparation for the quiz. Most will have forgotten the formula a week, a month, or a year later. This is because they will not use the formula again, unless they take calculus many years hence. In any case, whether they remember it or not, doing the exercises does not help them understand the formula.

Approach 2

Students are given a sheet of paper with a few copies of a certain trapezoid on it, with all the measurements indicated, including the bases, the legs, and the height. They are not given a formula. They are asked to find the trapezoid’s area. They are allowed to use scissors to cut out the trapezoids, and if they want to, cut one of them into smaller pieces that can be rearranged. Rearranging would allow them to use formulas they already know, such as the one for the area of a rectangle, a parallelogram, or a triangle. Students will almost certainly come up with different strategies.

(In fact, see how many strategies you can find.)

Students who find one quickly can be encouraged to look for more. Some students may not like the activity, because they are not told exactly what to do. The teacher can offer hints to them, or encourage them to get help from neighbors.

Once some strategies have been found, the teacher can lead a discussion where students demonstrate their approaches. All strategies will reveal that the lengths of the legs do not contribute to the final answer. In fact all strategies will yield the same answer for the area. A general formula can be the final punch line: applying any of the strategies to a generic trapezoid always yields the same formula. 

(Scissors are not absolutely necessary. For example, the activity can be carried out on paper, without any cutting. Whether that is preferable will depend on the specifics of a given class.) 


Even if the first approach includes a brilliant teacher explanation of the formula, I claim that the second approach is preferable. Many students who cannot remember the formula at some point in the future will be able to use one of the strategies that came up in the course of the exploration, either to find a particular trapezoid’s area, or to reconstruct the formula. This approach also carries the message that formulas can make sense, that there are many ways to solve a given problem, and that not everything needs to be memorized. A perhaps unexpected bonus is that the different solutions to this essentially geometric problem yield different interpretations of the formula, and some apparently different but actually equivalent formulas. Discussing this can help improve symbol sense. Finally, if the teacher has an excellent explanation of the formula that was not found by the students, nothing prevents him or her from sharing it. Starting with the hands-on activity does not prevent the teacher from offering an explanation, but it does mean that more students will understand the explanation.

Monday, January 23, 2017

Algebra Manipulatives

 A middle school teacher writes:
Just a little note and question about Lab Gear. I have been having so much fun with my students using Lab Gear again this year. The 3D-ness of it totally blows the other (cheaper) algebra tiles that I used last year out of the water!
I have heard this often from people who have used both tiles and Lab Gear. For one thing, the 3D-ness allows the blocks to represent monomials such as x^3, xy^2, and so on.

But besides the mathematical arguments, the blocks are easier to use, and more fun than the tiles.

Not related to the 3D-ness is another design feature that makes the Lab Gear work remarkably well: the 5, 25, 5x, and 5y blocks make it possible to quickly build relatively large products such as (x+5)^2 or (y+7)^2. Such problems with tiles take a lot of tiles and a lot of time. Moreover, the corner piece helps separate the length and width from the area:

(Readers who are not familiar with the Lab Gear can get information on my Web site. For a full comparison of all the algebra manipulatives, see this page.)
I gave the 7th graders an assignment to build their own Lab Gear Perimeter puzzle for homework and the results were incredible - it was so cool to see how deeply they understood the idea of length and perimeter.
I love that you asked your students to create their own puzzles!

Perimeter puzzles are a genre I pioneered im the Lab Gear books. They make a nice algebra-geometry connection, as they motivate combining like terms in a context where it makes sense, and where students have a need for that simplification. The 3D-ness of the blocks makes it possible to extend this to surface area problems and puzzles.
Although the 3D-ness also means kids try to make elaborate structures which topple to the ground! My threats to make them find the surface area of anything they build are met with "yay! we will!"
:) A good problem to have.
I noticed in the book published by Didax that you (they?) put the chapter on minus after the chapters on multiplying with Lab Gear. Is there a reason why? I have always done distributing the minus sign right after combining like terms.
That was my choice for the Algebra Lab Gear: Algebra 1 book. In the middle school book (Basic Algebra) I deal with minus at length, and early on, as it's a big middle school topic. That's probably what would make sense with your 7th graders.

In the Algebra 1 book, I use what I think is the best sequence for high school. Dwelling on minus early in a high school algebra class is not a good idea -- boring for some kids, confusing for others. Better get into topics such as factoring and distributing as early as possible, and save minus for later, at least in the context of manipulatives. Minus in the corner piece is complicated and not a good idea early on. Also, as far as I'm concerned, using the Lab Gear for equation solving is definitely not the main or first use of the blocks in Algebra 1.

That said, I realize that lots of people do Algebra 1 in middle school, which is why I recommend having both books and deciding on your own priorities and sequencing. You'll also have more examples for the most important topics, as those appear in both books.

Thanks for writing!