Tuesday, February 11, 2020

What is 8?

I've had a chance to spend more time doing math with the kids again and am hoping to write up our activities more consistently.  Let's see how this works out!

Graham Fletcher created a set of  Progressions videos for various elementary school themes. J3 and I recently went back to his page and found he had a new(er than we knew) progression on early number and counting.  Even for this simple topic, the video highlights some points we hadn't considered explicitly, for example distinguishing producers (of a number) and counters. Also, the cardinality point that smaller natural numbers are nested within larger numbers wasn't something we had talked about, but we soon realized it was part of many examples in how we understand numbers.

With that as inspiration, J3 and I decided to search for a range of examples of a single number, we chose 8, in different forms.  There is at least one obvious version we're missing.

Add a comment (with picture, if you can) to show other forms of the number 8!

Marking 8 on the 100 board, an easy place to start:

8 beads on the abacus shows the relationships 3+5 = 8 and 10-2 = 8 (also 100- 92 = 8)

8 can hide in plain sight. Without labeling the three lengths, it would have been hard to recognize the longer one as 8 cm and, for you at home, impossible to know without reference to show the scale.

It happened that, within the precision of our scale, two chocolate wrapped chocolate bars were 8 oz (2x3.5 oz of chocolate + about half an ounce of wrapping for each):

8 cups of water ended up being a lot, so this version unintentionally revealed a relationship 4 + 2 + 2 = 8

Though I'm not sure I can articulate why or show supporting research, I feel it is very valuable to build experience with physical models of numbers to create familiarity and intuition about what they are/mean. In particular, I hope this helped J3 anchor the importance of units of measure and scale in the interpretation of numbers.

Finally, this construction has nothing to do with the number 8 (or does it???)

Thursday, May 30, 2019

Quick 2019 resources for parents

A quick list of resources for an elementary school parent

Resources depending on prep time:
  • Grab-and-go

  • More prep time, in increasing order of advanced time required
    • http://mathpickle.com/: the puzzles and games are very good. 
    • Math Teachers at Play blog carnivals: https://denisegaskins.com/mtap/.  Variable amounts of prep time, but usually there’s at least one activity that is ripe for exploration, may take a bit of reading through the carnival to find a suitable one.
    • Mike's Lawler's blog: wonderful collection of (mostly) videos of his family working through problems, puzzles and mathematical explorations. Because his kids are older, it will take a little time to find something you think is suited for your son and then a bit to organize the activity.
    • Georgia State math standards: Despite the name "standards," these documents have a full curriculum with a collection of really great activities. As with any full curriculum, not everything is a complete winner, but there are enough gems. Also, this is probably the best resource for finding material to complement a kid's weaknesses.
This page is still the most comprehensive list of our favorite resources: http://3jlearneng.blogspot.com/p/favorite-educational-resources.html
Unfortunately, it is a little dated as I haven’t really been maintaining this blog in the last 2 years.

Friday, August 3, 2018

A context investigation

Note: I drafted this a while ago and never finalized the post. Reading it again, it seems fine and maybe interesting without additional work, so I'm publishing it.

Fake Math Models

Robert Kaplinsky wrote a note recently discussing fake math models and unnecessary context. This prompted an activity with the kids.

This issue seems to have come up a lot recently, so I've noticed a pattern: I really hate bad contexts.

Robert wrote: "it looks like the context is completely unnecessary to do all of the problems."
I would go farther: this context is harmful. The context creates a conflict between the specific new material (rational vs irrational numbers) and other important concepts (measurement and measurement error). Subtly, we are discouraging students from
(a) forming connections across topics. For my taste, surprising connections has to be one of the most beautiful and delightful aspects of math.
(b) using all of their ideas and creativity to understand a challenge.
(c) putting new mathematical ideas into a broader mathematical context (maybe I'm just repeating point a?)

I admit that the example only touches on these points lightly, but I suspect the accumulated weight over the course of a school math education is substantial.

If I were full-time in a classroom with a textbook, I'd be tempted to use it as follows:
1. create censored versions of all problems and examples (as you did)
2. work through the questions with the kids
3. Ask them what context they think the publishers originally included and why
4. show the published version
5. discuss (does the published version relate to the math, does it help them understand, does it add confusion, does it conflict with something they know, etc)

Tuesday, July 24, 2018

Playful Math Education Carnival #119

Welcome to Playful Math Education Carnival #119! Just to be clear, that exclamation is to express excitement, not factorial. Fortunately, you will have a bit of time before there's any danger of confusing this post with the edition (119 factorial).

Anyway, only the very coolest folks get to handle a MTaP edition that can be written with a factorial. And I just realized how close (and yet how far) I was to such glory.

119 Fun facts

  • 119 is the number to call for emergency services... in parts of Asia (wiki reference).
  • Of course, 119 backwards is 911 which is the US emergency services phone number
  • 119 is aspiring, the sequence formed by summing proper factors ends with a perfect number.
  • 119 isn't prime, but it almost feels like it
  • 119 = 7 x 17. I don't think products of consecutive primes ending in 7 has a name, but maybe it should?
Do you have other fun facts about 119? Please? Please?


I'm saddened to note the passing of  Alexander Bogomolny this month, and I dedicate the edition of the  carnival to him. The material he developed and made available on his site
https://www.cut-the-knot.org/ is truly amazing and remains with us for our benefit.


Aperiodical has an article from Benjamin Leis on the Big Internet Math-off.
Something James Propp wrote as part of the Big Internet Math-off: A pair of shorts


I was reminded of the game of Chomp! in Shecky Riemann's linkfest (most of which isn't elementary level, but worth investigating).

Cathy O'Neil tells her mathematician origin story. I hope all our kids can have an empowering math experience like this.

Discussion of a "square dancing" puzzle from Mike Lawler: part 1 and part 2. I think there is a lot more to explore here and hope some of you will write parts 3 and beyond... 

I always love game discussions. Set is a game you probably all know, but in case you don't here's an intro and a deeper analysis in the Aperiodical.

Pat Ballew writes about divisibility rules. Pat also discusses a fun XKCD in prime time fun.
I'm delighted at how this starts with something many take for granted (12 hour vs 24 hour time of day conventions) and then builds a fun exploration.

Middle school

Have you been waiting for someone to write the perfect post giving you an introduction to tons of Desmos activities? Well, Mary Bourrasa has done it for you.

Michael Pershan tweeted a pointer to a nice collection of logic puzzles on puzzling stackexchange.

Denise Gaskins pointed out a past note about factor trees and some cute wordplay from Danica McKellar's book: prime numbers are like monkeys.

This segues directly into a review of two number theory books by Ben Leis (also the author of the Big Internet Math off post above) in which he discusses some other visualizations beyond factor trees: 

High school

Ben Orlin invents and illustrates a new adage that there are no puddles in mathematics, only oceans in disguise

More advanced

Mathematical theorems you had no idea existed because they are false: https://www.facebook.com/BestTheorems/
Have fun finding counterexamples. Also, link disproves the conjecture that there is nothing worthwhile on facebook.

The Scientific American Blog has been running these columns on "my favorite theorem." Go back and take a look (I think this was their first one): Amie Wilkinson's favorite theorem.

A fascinating discussion of the Fields' Medal and some ideas about what it should be supporting. 

What was the score? Maybe the sum of scores was 119?

Wednesday, May 30, 2018

Ambiguity in math class

Math class is a special place. We've talked before about some of the special assumptions that are based into that context: teachers pose questions, students answer questions, all questions have answers, questions include all the necessary information, answers are usually "nice," problems can be answered with the tools students have (just been) taught, diagrams are indicative while the underlying true forms are perfect, etc.

Of course, not all math classes make these assumptions or leave them implicit, or are constant about which ones are in force, etc.

In this post, I want to pick up a thread related to the "one true answer" myth: problems that have multiple interpretations.

You are driving from your house to a soccer tournament. The distance is 120 miles. For half of the trip, you drive 60 mph. For the other half, you drive 30 mph. What is your average speed over the whole drive?

Where's the ambiguity?
For the teacher who poses this problem, there is no confusion. Obviously, students are meant to calculate that it takes 1 hour to drive the first 60 miles and 2 hours for the second 60 miles. That means it took 3 hours for 120 miles, or 40 mph average speed.

The catch: what does "half of the trip" mean? As an alternative, it could mean half the time of the drive. If that feels contrived, consider the following natural statements about travel measured in time instead of distance:

  • "The drive took 3 hours; we stopped for a snack half-way." In this case, time and distance are equally natural in normal conversation.
  • "The flight took 6 hours;  I read half the time and slept the rest." In this case, time is the more common metric, but it wouldn't be considered unusual for someone to talk about the distance they flew.
  • "We were gone for 2 weeks, half at the beach, half visiting our cousins." Here, time is the natural metric, while it would seem strange to focus on distance. However, a vacation spent hiking the Appalachian trail or cycling across country would shift the balance back to distance.

Sources of ambiguity

I came up with four potential sources of ambiguity in math questions:

Things that can be measured in multiple ways. 

This extends the idea from “half a trip” ambiguity about distance or time. J1 and I had a discussion a couple of weeks ago where we measured chocolate bars and cookies using three different metrics: mass, cost, utils. For example, which is more:

  • 100 grams of chocolate that costs $2.00 and you value at 100 utils
  • 80 grams of fresh baked sugar cookie that costs $2.50 and you value at 90 utils

In business, it is common to have to deal with the ambiguity of whether “stuff” is measured in physical amounts or monetary value.

Pronoun ambiguity

For example: Ellis had 10 strawberries. Ellis gave 4 to his father and he ate 2. How many does he have now?

Who is meant by each occurrence of "he"? In each case, it could mean either Ellis or the father which leads to 3 distinct answers: 6, 4, or 2.

I accept that this is an example of bad English, but we're in math class and never claimed to be masters of language (did we?)

Tense ambiguity

In the prior story it could be that giving the strawberries away and eating them happened before the state where Ellis had 10. Let's add some extra story context to make this alternative more clear:
Ellis still had 10 strawberries. He bought a pack of 16, but Ellis gave 4 to his father and he ate 2.

I think this alternative interpretation is more of a stretch, but I've seen cases where the uncertainty about when things were happening is more natural.   

Assumption of scalability

Joe can bake 2 cookies in 20 minutes. How long does it take him to bake 4 cookies? 400 cookies? 4 million cookies? 4 quadrillion cookies?

I saw one math class question that involved writing books, a task which is very unlikely to happen at a constant rate.

Your challenges

  1. Find other sources of ambiguity that can infect (or add spice to) math class problems.
  2. For N a positive integer, create a puzzle that has N distinct solutions based on (reasonable) alternative interpretations.

Tuesday, February 20, 2018

What is your function? More excuses to delay bedtime

J1 (5th grader, looking for an excuse to stay up): What are you working on?

J0: I'm writing a review of a book.

J1: The one we got from math circle (Martin Gardner's Perplexing Puzzlers and Tantalizing Teasers)? 

J0: No, the one about Funvillians.

From Natural Math!

J1: Tell them that it was fun!

J0: You really enjoyed reading it. I'll make sure to mention that. I was thinking that we should have used it as an inspiration to make our own adventures.

J1: You mean, like creating new characters with their own powers? We could have heroes who control fire and ice, some others that can go forward and backward in time.

J0: Is that how the Funvillian powers worked? I thought they needed to have inputs. For example Marge's power only works on two exactly identical objects.

J1: Sure, the current time is an input and the output is the time in 5 minutes. Or another one can do the reverse.

[pause, maybe he's starting to go to sleep?]
J1: Or... maybe we could make up some adventures where the Funvillians from the story have to solve their own challenges.  They could meet some villains... not Villians! (laughing)

J1: The one who can duplicate things ... what if that power could be used on people? After they were copied, would they all have to do the same things? For example, if I were copied and I raised my arm, would the other one have to raise his arm, too? Could they think different thoughts?

J0: Well, when they copied two toys, they could play with the toys separately. The toys didn't have to do identical things.

J1: Oh! But what if they were changed a tiny amount? Would they still be considered identical and could they get reduced down to one copy?

J0: I don't know. Where do you think the powers come from?

J1: maybe from living in their magical land. Probably when they have spent enough time there, a power develops.

J0: There, so that's what I'm going to write about. Thanks!

J1: Remember to tell them it was fun!

On Fridays for the last several months, my fifth grader and I have been spending 2 hours in the evening doing math together. By that time of the week, I'm not always feeling energetic enough to properly plan an activity or exploration. Looking to give myself a break, last week, I brought Sasha Fradkin's book Funville Adventures for J1 to read during the session.

He was engrossed and finished it with some amount of time to spare. Maybe 90 minutes of reading, leaving us 30 minutes to discuss.  He had read the addendum, so was already primed for talking about functions. In addition, he still remembered past conversations about "function machines" and programming functions. Using the characters as references, though, he found it much more intuitive to understand invertible and non-invertible functions. We talked about examples of arithmetic functions that were similar to different characters' powers and had fun giving examples of what would happen if different characters used their powers in succession.

The experience, so far, suggests that this is a helpful model for understanding functions, more human and vivid than what we'd previously done with function machines.

And remember, it was fun!
(now go to bed!)

Tuesday, October 24, 2017

Math Teachers at Play Blog Carnival #113

Welcome everyone to the 113th Math Teachers at Play blog carnival! As usual, I'm lucky to have the best month to curate. 113 is the prime MTaP because:
  • 113 is prime
  • all permutations of the digits are prime
  • all 2 digit subsets of the digits are prime
  • the product of the digits is prime
  • the sum of the digits is prime
  • 113(4) (113 in base 4, the smallest base that is sensible) is also prime!
  • 2113 - 2 is divisible by 113. Wow! (see Jordan Ellenberg's Favorite Theorem)

October is for Play

October is a perfect month to talk about mathy play because of:

Later this week is the largest international games convention in Essen, Germany. I'm jealous of any of you who get to go. For the rest of us, I've collected a bunch of great math games and explorations to keep us happy.

Before we move off Spiel, though, take a look at the logo above again.
What do you notice?
What do you wonder?

Click this button to compare with another version of the logo:


In the spirit of Malke Rosenfeld's Math in Your Feet, Mrs. Miracle's post about beat passing games can inspire a whole-body exploration of patterns. I like expanding beyond visual patterns and the fact even very small children can create their own beat pattern.

For some reason, this old Christopher Danielson post resurfaced on my RSS reader. While it is an old one, I hadn't seen it before, so maybe you missed it too or will appreciate reading it again: Armholes. Maybe it is easier to be patient while waiting for the kids to get dressed if we are also exploring math at the same time?

How many holes?

An online math competition for elementary kids: BRICS Math. I like using math competition questions as a jumping off point for further conversations and explorations. Sometimes the questions have natural extensions (what if we changed this number?) and other times we just talk about what the kids found interesting about the question or what it made them think about.

Iva Sallay (who has hosted the last edition of MTaP) makes a Halloween 10 Frame (just in time!)

Here are some wonderful images and gifs from Gábor Damásdi. They could be a good prompt for Notice & wonder for young kids and older ones:

AO Fradkin talks about a tricky game that helps develop mathematical language: A figure with pointy things...
This question: why do we bother with defined terms and mathematical language fits nicely with Chasing Number Sense's exploration of the definition of a polygon: Polygon is a shape that is really big.

Denise Gaskins, the wonderful unifying force behind this blog carnival, reminds us of 30+ things to do with a 100 chart. I have one more to add: our family first learned to play Go on a 100 chart with some small blue cubes and bananagram tiles, before we had a chance to buy our first dedicated board. Here's an old snap from the beginning of the year:

Middle School

If you like chained fraction puzzles (we do!) and you like thinking about concrete manipulatives (we do!!) then you'll enjoy this post from Bridget Dunbar: Thinking in th Concrete.

Another post from Iva Sallay uses candy to teach equation solving: Solve for X with candy. With Iva's help, we're certainly ready for a mathy Halloween.

Presh Talwalkar at Mind Your Decisions occasionally posts viral puzzles with some nice explanations. I enjoyed this one (octagon in a paralellogram) because it fit with a problem solving strategy we've been practicing recently: test a special case. Here, we tried starting with a square, then discussed whether that was really a "special case" or fit the general situation.

Mike Lawler, as usual, has some fun posts, this time I picked out his videos talking about Tim Gowers's intransitive dice.

Different from Tim Gowers's dice?

Huge jars of coins are wonderful, for so many reasons. Kristen (Mind of an April Fool) shares a fun 3-act lesson: Sassy Cents. Our family has gotten a lot of mileage out of doing notice and wonder at home with similar 3-act lessons.

Jim Propp contributed to the excitement of Global Math Week with a sort of History of Exploding Dots. I have an especially warm feeling for this story because he includes mention of the "minicomputer" idea created by Frederique Papy. These minicomputers figured prominently in my own elementary math education.

High School/More advanced

Continuing with the theme above around language, definitions, precision and math, Mr Orr gives us 3 Desmos Activities for Talkers & Drawers.

Curiousa Mathematica shares a Putnam exam question that is actually very accessible: Spots on a ball. Try to think about it before reading the solution.

Patrick Honner talks about some of the math related to gerrymandering in Wasted Votes. In his discussion, he describes a game that sounds very much like Mathpickle's A Little Bit of Aggression (pdf here.)

SolveMyMaths has done a series this month on trig identities. These build step-by-step pictures to help understand what is going on, for example, in the angle addition formulas. Take a look (they are easier to understand than this final step picture, but it is one of my favorites):

Teaching Resources

For all of us who sometimes have to find a math curriculum for our kids, David Wees has created his checklist of necessary characteristics: Questions about Curriculum.

What makes a good school? Jane Mouse (in russian) explains that there is no "best" school. For those of us who teach our own kids, one interpretation is that we should try to expose the kids to a variety of modes and styles. Also, what is working now might change over time.

Sam Shah offers a number of hacks for making your own material. My personal recommendation is for you to try this out with your kids: make problems together and discuss the process. What makes a good question? What makes a hard vs an easy question? Can you create problems with only one, more than one, or no answers?

Resourceaholic (Jo) has, you guessed it, a presentation on resources: Power of Six presentation. Be sure to take a look at Jo's Resource Library for ideas when you need secondary school material.

I'm sure there are a lot of other great posts with families and teachers sharing their math games and explorations. Please add comments to let me know about your favorites from the month (or older ones)!