Sunday, December 31, 2017

Arithmetic, by Paul Lockhart

This book is part history and part philosophy of arithmetic. He also includes a few exercises in italics (one every few pages).

I love his easygoing tone (arithmetic is not mathematics, it's an art or craft, something like knitting, I enjoy it, and I hope you will), his low-key sense of humor, and his perspective.

I loved his book Measurement, and look forward to someday working through all the challenging problems in it. This one is much easier, and yet it's not boring for me.

It would be a lovely book to read to kids and think through together.

(It came out this year in hardcover. $22.95. I love the cover, though I know I shouldn't judge a book by that...)

[Edited to add:
I wrote this review before finishing the book, because I was so excited. I've read a few more pages now. Sadly, Lockhart is sometimes sexist. On page 45 he mentions the (Japanese) emperor's concubines, and says "Now, this is why people do arithmetic!" (To please the concubines.) No. It's not. And I thought better of you, Paul Lockhart.

I still think I'm going to love 99% of this book...]

Saturday, November 4, 2017

Re-reading Archimedes Codex

I am re-reading The Archimedes Codex. It is fabulous. A detective story of history, science, and math. (I wrote my first review of it two years ago.)

Both authors have a sweet nerdy guy sense of humor, gently self-deprecating, piercing when it should be.
"What are readers today [of science] afraid of? They are afraid of equations. With good reason: they were force-fed such equations for several, terrible years of their childhood and adolescence." 

I actually think he (Netz) is wrong here. I love math, and I love the beauty of some equations, but equations are still intimidating when you don't yet see what they're saying.

Here's another math-related quote, pointing in a different direction:
"Archimedes wrote out this problem in verse! A poet-mathematician! - the thought seems to us absurd, but it was natural for Archimedes, whose entire science was built on a sense of play and beauty, on hidden meanings."
Not surprisingly, I disagree with him here too. Some of us like mixing poetry and math. And the problem written up this way was a silly one. Archimedes was kind of like Martin Gardner with a few of his problems, and it makes sense he'd be playful in his presentation of it. Kind of like this one by Mike Shenk (sorry for the xmas reference this early).

Monday, August 14, 2017

First Day, Again

First day of classes. I was not as excited as usual. But I had my prep done, and once I got in the classroom, I loved it like always.

25 people in beginning algebra. I tell them how math is not about memorizing but about making sense. I get them talking in groups about the first time math didn’t make sense and got past them.

I ask them to estimate the percentage of the population that’s uncomfortable with math. First one says 100%. I almost laugh, but manage not to. Percentages all over, 32% up to another at 95%. I tell them I don’t know either but I guess 70 to 80%. So that means most elementary teachers are nervous when they teach fractions, and then they pass it on. I think I see a few nods.

I ask how to make meaning of 3 -5. Someone says you go past 0. I say “You’re talking about a number line, right?” And I draw it. I say I like that, but how can we give some real world meaning to this problem. Someone else says “Debt”, but I hear “Death” at first. I shake my head at my bad hearing. (I hope it doesn’t interfere with my teaching.) And I flesh it out. “Yeah, you’ve got $3 in your pocket. And you want to buy a $5 something. What does the 3 -5 tell you?” A student says, “How much you need to borrow.” I add in the temperature model, which I tell them might have worked better for my students back in Michigan who had experienced 3 degrees, and going down 5 degrees from there.

Then I give them 31 – 52. They discuss in their groups. It’s easier than I meant it to be.

We discuss the syllabus in between other things. I give them a sheet full of magic squares that use negative numbers. Some use fractions. I’ll need to check in tomorrow to see how they did with those.

I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever. That only happens about a third of the time with this course. I can hope…

Then I had to run to my next class. Statistics. I had them average the ages of 5 other students, and type their average onto my computer. I averaged their averages. Which is not the same as just averaging everyone. But it often comes close. Then I put a number line up, and we each put a dot at our age. That’s a dot plot. I showed how it’s skewed right. And talked about how median is a better way to show the center than mean (which is the average they had done).

I had an hour to eat lunch and chill.

Then I taught calculus. I love that class! Every time. Draw y = x squared. Draw a tangent line at x=3. Estimate the slope of the tangent line. What makes this different from algebra is that we need the idea of infinity here. Ahh… Happy me.

Friday, June 30, 2017

Math Teachers at Play #109 (a blog carnival)

Who is 109?     109 is a twin prime, twinned with 107.    (from

+ If 109 is written in Roman notation (CIX), then it becomes reflectable along the line it is written on.
+ The pipe organ at the Cathedral of Notre Dame in Paris has 109 stops.
+ When chilled below minus 109°F, CO2 becomes a solid, called dry ice.
+ 109 equals the square root of 11881 or 118 - 8 - 1.
+ The only three-digit prime formed by concatenation of consecutive numbers. [Silva]
+ 109 = 1*2+3*4+5*6+7*8+9. [Silva
+ The Sun is just over 109 times the diameter of the Earth. [Friedman]  


A Puzzle: Can you make 109 from four 4's? (I don't promise that it's possible...)

At age 109, Augusta Bunge became the youngest living great great great great grandmother. Is that mom to the fifth power?

Math Teachers at Play is a monthly blog carnival, hosted at a different blog each month. I was hoping to give you 109 math links this month, but life intervened (parenting...) long before I got through my storehouse of cool stuff. There are plenty of goodies here, but not as many as I'd hoped.

There has been an explosion of super cool mathy books since I last hosted MTAP. Here are some I know about. I am embarrassed to admit that I haven't read most of them, and so I can't guarantee how cool they are. Let me know in the comments.


Puzzles & Games 

Early Math


Probability & Statistics

Writing in Math Class 

Math and ...

On Teaching 

Random Stuff

I have more but it's bedtime and June is ending. Would you like to see your favorite blog post in next month’s playful math blog carnival? Submissions are always open!

(Note: Edited on 7/1 to add a few forgotten links, and fix a few broken links.)

Tuesday, June 6, 2017

What I learned at CAP's Community of Practice

CAP is California Acceleration Project. Check out their publications. The first time I attended one of their conferences, I struggled with the word acceleration. It does not mean getting through the material faster. It means getting to the good stuff faster - shortening the pathway of required prerequisite courses students must take before taking a college level course. Their work is mainly with math and English, the two subjects that generally hold students back.

In math, the college level course for someone not interested in STEM is statistics. Students take a placement test, and the large majority (86% at my college) are placed in remedial courses, anywhere from 1 to 4 levels below the statistics course. Imagine a student starting 3 levels below, at pre-algebra, which is where over half of our students are put by the placement test. If we had phenomenal success rates, with 90% passing each course, and phenomenal persistence rates, with 90% going on to the next course, we'd still only get 43% of these students finishing statistics (.9^8 = .43). What happens to the other 57%? Usually they give up on college, for at least a while.

Because housing is pretty segregated in the U.S., and that makes k12 education pretty segregated, with people of color getting less resources dedicated to their schools, this becomes a civil rights issue. CAP is dedicated to:  changing the way we place students (many who do badly on the placement test can still pass a college statistics course), developing models for co-requisite courses that students can take with statistics to improve their success rates, and developing radically shortened and improved remedial pathways (creating a pre-statistics course that prepares students with just enough algebra and lots of data analysis).

I have been attending their workshops whenever I can for the past few years. This past weekend I went with two other math faculty and 5 English faculty. Even though I've seen much of the information before, I still got a lot out of it. (Maybe I'm a slow learner!)

Here's something I put together yesterday at the request of our dean for equity, which summarizes some of the important points I learned...

Planning a High-Impact Course

More important than any one course are these 3 principles:
  • Create separate pathways for STEM and non-STEM.
  • Place students as high in the sequence as possible.
  • Shorten the sequence as much as possible.

CAP’s 5 design principles
  1. Backward design
  2. Low-stakes, collaborative practice
  3. Relevant, thinking-oriented curriculum
  4. Just-in-time remediation
  5. Intentional support for affective needs

High Performing Math Classrooms (Internationally)
James Stigler on high performing math countries.
All have these things in common:
  • Productive struggle
  • Explicit connections
  • Deliberate practice, increasing variation and complexity over time

Lesson Planning (CAP)
Given a topic you want students to learn through groupwork,
  • Identify the prerequisite skills needed,
  • Decide whether these will be addressed through productive struggle (ie not addressed overtly), targeted group activity, or just-in-time mini-lecture, and how you’ll do that,
  • Plan main activity,
  • Plan closure (vital for making explicit connections)
  • Note: Over-scaffolding brings down the thinking level required.
  • (Sue has a form from CAP. If this link works, it’s to all the CAP materials: )

Thinking Levels
The thinking levels mentioned above come from a study by Quasar. Here’s the relevant info:
“This research yielded two major findings: (1) mathematical tasks with high-level cognitive demands were the most difficult to implement well, frequently being transformed into less-demanding tasks during instruction; and (2) student learning gains were greatest in classrooms in which instructional tasks consistently encouraged high-level student thinking and reasoning and least in classrooms in which instructional tasks were consistently procedural in nature.” (Stein p. 4)

QUASAR Task Analysis Guide (adjusted slightly to address statistical thinking as well as mathematical thinking)
Lower-Level Cognitive Demand
Memorization Tasks  
Involve either reproducing previously learned fact, rules, formula, or definitions;  
Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure;  
Not ambiguous; clear and direct instructions to reproduce previous material;  
No connection to the concepts or meaning that underlie the fact, rules, formula, or definitions.

Procedures Without Connections Tasks  
Algorithmic; direct instructions to use a procedure or the use of the procedure is evident based on prior instruction, experience, or placement of the task.  
Require limited cognitive demand for successful completion.
There is little ambiguity about what needs to be done and how to do it.  
No connections to concepts or meaning that underlie the procedure being used.  
Focused on correct answers rather than developing mathematical or statistical understanding.  
Require no explanations, but may require students to “show work”.

Higher-Level Cognitive Demand
Procedures With Connections Tasks  
Focus students’ attention on the use of procedures or concepts for the purpose of developing deeper levels of understanding of mathematical or statistical concepts and ideas.  
Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.  
Usually are represented in multiple ways (e.g. graphs, tables, numerical summaries, verbal descriptions).
Making connections among multiple representations helps to develop meaning.  
Require some degree of cognitive effort.
Although general procedures may be followed, they cannot be followed mindlessly.
Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

Doing–Mathematics or Doing–Statistics Tasks  
Require complex and non-algorithmic thinking (i.e. there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example).  
Require students to explore and understand the nature of mathematical or statistical concepts, processes, or relationships.  
Demand self-monitoring or self-regulation of one’s own cognitive processes.  
Require students to access and make appropriate use of relevant knowledge and experiences  
Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.  
Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.

Saturday, March 4, 2017

At the Julia Robinson Math Festival Today

Julia Robinson Math Festivals invite kids to play with math puzzles that start easy and offer harder questions as you go along. Today's festival was at Bentley School in Lafayette. (Some festivals are open to the public, and are much bigger.)

I was working the Pilgrim's Puzzle table. We had this puzzle to work on.

It was really fun watching kids and parents get engaged with it. Some paths give you fractions, and then taking away 2 can give you something like 1/8 - 2, which can be pretty confusing for a 3rd grader.

The first time I tried to help a kid with a problem like that I was not able to find an image that made this sensible. When B was stuck with a problem like this, I came up with anti-matter apples. It worked! We imagined 1/8th of an apple, and imagined two anti-matter apples. We cut the 2nd one into 8 pieces, took one of those pieces and exploded it with the regular 1/8th slice to make a poof and then nothing. So we had one anti-matter apple and ... 7/8ths of another, which we wrote as -1 7/8. Done.

I will be teaching beginning algebra in the fall. I don't think I've ever found an image for negative fractions that worked as well as I think this one will. I'm excited.

Here's B and me.

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