My Math Education Blog

"There is no one way"

Monday, January 23, 2017

Algebra Manipulatives

 A correspondent 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.
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, and that teachers may not agree with my sequencing, 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!


Tuesday, January 3, 2017

Yet More on Homework

 A correspondent writes, presumably in response to my lagging homework concept:
I love your approach and a lot of the details. I guess my only immediate reservation would be the necessity for homework. A lot of U.S. kids simply can not do homework.  They are homeless; have full-time jobs; are tending ailing grandparents or little brothers, etc. 
That is an important point, so I should clarify my views. Homework is not a matter of principle for me. It is a component in a comprehensive approach, and the same goals can be accomplished in other ways. In my teaching, it played a key role, because most work in class was collaborative, but almost all assessments were individual. As one of my colleagues put it: "We work in groups today in order to learn how to work alone tomorrow." The process was (more or less):
  • Week 1: You learn a new concept in class, where you can get help from your peers and the teacher.
  • Week 2: You apply the concept in short homework assignments. Those help you sort out what you understand from what you are still working on, which will make it easier for you to get exactly the help you need when going over the homework with your group.
The message is: you need to know how to do this on your own, but you can get as much help as you need. You are not being rushed or pressured. My goal as the teacher is not to separate those who can from those who can't: it is to get to where everyone can. (To make this message even clearer, the quiz on this material is not happening until Week 3, and you'll have a week to turn in quiz corrections in Week 4.)

If a student cannot do homework because of their home situation, the school should offer study hall time so that "homework" can be done at school. If the school cannot or will not do that, and many students are unable to do homework, then some class time can be allotted for that: time where you work on your own, after plenty of time working with peers, and prior to being quizzed. Of course, giving up class time for this will reduce coverage, but it may be a price worth paying.

However, I should say that I have heard from more than one teacher that homework completion increased dramatically once homework was lagged. It turned out that the reason many students weren't doing the homework was because they didn't know how to do it. Lagging gave them time to learn the concepts, and made homework possible. So before abolishing homework, I would try both lagging it, and offering in-school time for it. More students will learn more math than with no homework at all.

See below my signature for an excerpt from what I wrote in 2013 in response to someone who asked for my views on abolishing homework. Today's post is intended to complement what I wrote then — I still agree with it.

-- Henri


Most learning happens in class, and one should not overdo homework. Too much homework only antagonizes kids and in most cases, it does not help their learning.

On the other hand, a small amount of homework is a good thing:

  • It is a form of differentiation, as it allows kids to take different amounts of time to do the same assignment. (The nature of what happens in class in a cooperative learning culture is that racing is discouraged, and kids work more or less at the same pace, with the faster students slowing down to help others.)
  • It gets the message across that it takes work to learn anything substantial, and that while in class we work in groups, the ultimate goal is to understand the material well enough to deal with it on your own.
  • It's a place to do the often necessary but often boring work of basic drill and review, thereby saving class time for more interesting and substantial engagement.
In my class, a powerful argument in favor of homework came from the students themselves. In course evaluations, it was common for students to tell me that what helped them learn the most was going over the homework with their classmates. I usually allowed about 15 minutes for that at the beginning of class. During that time, I walked around and recorded a 0 (did not do it), 2 (great job), or 1 (somewhere in between). Students helping each other is far more efficient than me explaining things to the class, because it allows different tables to focus on the parts of the assignment they each need to focus on. True, there's a risk that all the students at a given table are doing something incorrectly, but that doesn't happen very often. Besides, after a quick homework check, I'm walking around, ready to intervene if I see this.


Monday, December 5, 2016

Time and Tide

This is my yearly report on the Asilomar conference of the California Math Council, Northern Section. Because I was presenting three times, I didn't end up attending as many sessions as I would have liked. As always at Asilomar, I enjoyed hanging out with my ex-colleagues, running into friends, and meeting the occasional fan of my Web site. Not to mention the chance to take a walk along the Pacific Ocean. (Thus "tide" in the title of this post.)

On Friday, I co-presented a session with Lew Douglas: A Deep Dive Intro Transformational Proof. You can see the description in my previous post. Lew and I are gradually building up a logical framework for a transformations-based high school geometry course, and working on this session pushed our project forward. You can see our latest writings on my Transformations page. The session went well, I believe, except that there was not enough time for most participants to do the GeoGebra activities we had planned.

On Saturday, I presented Computing Transformations Using Complex Numbers and Matrices. That session was based on my approach to complex numbers in Algebra 2, and on some of the material I developed for my Space course. It went reasonably well, judging by the evaluations, though I did not dwell sufficiently on the trickier ideas, and moreover I ran out of time at the end. You can find the relevant links on my Talks page.

In both sessions, time was a bit of an issue, as it always is for a new presentation. But time was actually the topic of my Ignite talk: Time Pressure: Bad for Students, Bad for Teachers. Ignite is a format where you have exactly five minutes, and your presentation includes exactly 20 slides. The slides succeed each other every 15 seconds, whether you're ready or not. Preparing for this was pretty stressful. You can watch it on YouTube, minus one slide which didn't make it into the edited version. Meanwhile, I'll summarize the key ideas here.

If we want to teach for understanding and retention, and if we want to reach a broad range of students, we need to reduce the tyranny of the calendar ("you must learn this by Friday!") and the tyranny of the clock ("you must finish in 5 minutes!") I have written a fair amount about the calendar already. [See for example Lagging Homework (my most popular blog post ever) and follow the links therein. Also: Pruning the Curriculum.] What about the clock? Here are some thoughts, at much greater length than was possible in the Ignite format.

Classroom Discourse

Even in a student-centered, collaborative classroom, teacher-led whole-class discussion is helpful to share ideas, to move the conversation to the next level, and to bring students into the institution of mathematics. The goal of classroom discourse is not to get through it as quickly as possible! Most of us tend to ask a question, pick the first student that volunteers an answer and move on to our next question. That approach is absolutely guaranteed to leave most students in the dust, and to limit participation to a handful. Instead, we need to find ways to slow down, so everyone has a chance to think and an opportunity to participate. Here are some techniques I've used to slow things down:
  • "I'm glad you're enthusiastic, but you should raise a quiet hand if you have an answer. Blurting it out deprives your classmates of a chance to think." Never call on a student who blurted.
  • Before choosing a student, wait until many hands are raised. You may even count. "Five hands... Seven hands..."
  • Ask students to give their answers to a neighbor, before choosing someone to speak to the class. This dramatically increases participation, and emboldens some shy students.
  • Ask students to rephrase what a classmate has said. This is easier than coming up with an original idea. This technique makes it possible to call on more than one kid for an important question, and over time, it encourages students to listen to each other, not just to the teacher.
  • Write several answers on the board, keeping a poker face, and then discuss them. Or: have the students vote on which one makes sense to them, have the discussion, and then vote again.
  • If a student is stuck while speaking to the class, or makes a big mistake, give them the opportunity to choose a classmate to help them (among those who are raising a quiet hand, of course.) This helps take the sting out of being stuck or having made a mistake, as the student's attention turns to their newly acquired power.
  • Make mistakes that reflect student misconceptions as observed in this or previous classes, then discuss the mistakes. This models a healthy attitude towards making mistakes, which is a necessary part of learning.
If you do all these things, discourse will be slower, and conversations will be more useful to more students. (I have written about this in my cheerfully titled article and presentation: Nothing Works, where I present a wide range of teaching techniques, none of which will solve all your teaching problems.)


At most schools, math is mostly assessed by way of quizzes and tests. Unfortunately, those are usually conducted under tremendous time pressure. But why does it have to be so? What if your quizzes were shorter? What if you gave students as much extra time as they need? What difference does it make if they answer a question at 10:20 and not at 10:15? What if you used at-home assignments, with no clock pressure, as part of your assessment repertoire? What if those included opportunities to write quiz and test corrections?

This is an equity issue: many students tell us that they understand the math well, but that they freak out when under time pressure. It is only fair to give them a chance to demonstrate that by using a range of untimed assessments.

(Check out my series on assessment, starting here.)

Thanks for reading all the way to the end!


PS: The day before the conference, I did pentomino activities with an after-school math circle for children in grades 2-4. At one point, a girl who had done a great job solving puzzles (perhaps an 8-year-old?) asked her mother: "Is this math?" I interjected: "Yes!" Her response: "I thought math was supposed to be serious." I was happy that my workshop had challenged that misconception! And yes, pentominoes are great with all ages. See my Geometric Puzzles in the Classroom page.