Tuesday, February 18, 2020

Play Cribbage, Make 15s

I recommend this game to develop number sense using ten frames and have fun. Players find patterns and make 15 via addition. There is some beginning combinatorics when scoring 3 and 4 of a kind. (if I have three 2s, how many pairs do I have? What if I have four 2s?)

Rules for 2 players described below. (Rules for 3 players)

Cut to deal, low card is dealer. Face cards are 10, ace is 1.

The Players
    are dealt 4 cards each.

Pegging occurs.

Each player keeps 2 cards and puts 2 cards in the Crib.

Hands are scored and pegged. Scoring.

The Crib is scored and pegged.

Roles switch and repeat until someone pegs to finish.

#mathchat #elemmathchat #playmath

(image credit: https://images.squarespace-cdn.com/content/v1/54fa13a2e4b0a8165372cfc9/1541273685000-UD3A7WCDTLBUBH4SWIJN/ke17ZwdGBToddI8pDm48kDHPSfPanjkWqhH6pl6g5ph7gQa3H78H3Y0txjaiv_0fDoOvxcdMmMKkDsyUqMSsMWxHk725yiiHCCLfrh8O1z4YTzHvnKhyp6Da-NYroOW3ZGjoBKy3azqku80C789l0mwONMR1ELp49Lyc52iWr5dNb1QJw9casjKdtTg1_-y4jz4ptJBmI9gQmbjSQnNGng/P1050426.JPG?format=1500w)

Monday, July 29, 2019

Making Tens Matters; Play Shut the Box

Math Toys and Games for Young Learners

In this post I highlight toys, games and habits to develop young mathematicians.

Age 0-3

The concepts of cardinality (numbers have quantity) and ordinality (numbers have order) are not obvious. It is one thing to learn a string of words "one, two, three, four..." and another to understand that each of those words represents a quantity, or that the order of those quantities carries meaning. These concepts are reinforced through multiple representations of number and quantity. Have fun. Learn from mistakes. Don't punish children's partial understandings. Build upon them.


Count physical objects with your kids
    Ask "How Many?"
           "How do you know?"
           "How else do you know?"
Play with patterns
    Ask "What comes next?"
           "What else could come next?"

Physical Toys

Be careful of choking hazards.
Check out Frebel's Gifts as a progression of mathy toys to orient a child for future mathematics.

Digital Toys


Age 3-7

Making tens is silly; why wouldn't a young child to rebel against such a mundane task? Memorizing a 12x12 table of numbers is worse yet. Many games reinforce these foundational math facts without the pain of forced repetition.


Ask "How many?" "How far?" "How long?" "How much?"
       "What units?"
       "How do you know?"
       "How else do you know?"
       "What comes next?" 
       "What else could come next?"

Physical Toys

  • Shut the box
  • Board games (Sorry, Candyland, Monopoly, Careers, Katan)
  • Card games
  • Dominos
  • Backgammon
  • Chess
  • Go
  • Cribbage

Digital Toys

Number Sense
  • Dragonbox Numbers
  • Motion Math Zoom
  • Motion Math HD
  • Dragonbox Big Numbers
  • Wuzzit Trouble
Visual Spatial
  • .projekt

Age 8-12

Fractions are daunting. All of the symbols and numbers have new rules that are easily confused. Abstraction seems so irrelevant. How is the letter x going to help me in the real world? The symbol barrier can be overcome when learners are given time to mathematize a concrete experience and use symbols to represent mathematical features of their lived experience. Games like Slice Fractions and Dragonbox Algebra demonstrate this idea.


Cook together. Weigh and measure objects. Cut things in halves, thirds, quarters... Fold clothes together. Demonstrate that math is all around us, don't pressure children into seeing math as right and wrong; and the rules and symbols will come with time.

Digital Toys

  • Slice Fractions
  • Amplify Fractions
  • Refraction
Proportional Reasoning
  • Ratio Rancher
  • Dragonbox Algebra
  • Game over Gopher

In a previous post I discussed resources for high math achievers. That may be relevant for you.

For Teachers

What apps, websites, software programs, etc. do you use with students (or have them use independently) to improve their math skills? Why do you use these tools? What are their strengths and weaknesses?


  • Google – search in class with students to teach searching strategies
  • Wolfram Alpha – use to demonstrate function behavior
  • Wikipedia – search in class and discuss how to validate claims and track sources

Skill Practice

  • Open Middle - open questions provide more practice and engagement
  • Kahoot - gamified knowledge quizzes
  • Motion Math HD - Fractions
  • Motion Math Hungry Fish – Adding positive/negative Integers
  • Motion Math Zoom – Number Line and Place Value
  • Sumdog – basic skills + some metrics to demonstrate gains
  • ST Math – spatial reasoning
  • Khanacademy – Knowledge Graph with Demonstration and Practice
  • IXL - vast range of topics

Conceptual Understanding

  • Desmos - engaging classroom activities to promote mathematical discourse
  • Brilliant.org - engaging critical thinking tasks, sequenced into courses
  • Geometer’s Sketchpad/Geogebra - visualize/simulate mathematics
  • NLVM - virtual manipulatives
  • Shodor - interactive activities

Top 3-4 concepts/skills that hold students back

Algebraic expressions: Integers, Fractions, Order of Operations

Functions: Multiple Representations, Relational Reasoning, Inverse Functions

Proportional Reasoning: Scale Factors, Ratios, Percentage

Statistical Reasoning: Randomness, Normal Distribution, Conditional Probability

For more product reviews and tips for raising kids in our digital world, check out Common Sense Media.

This post is just a scratch at the surface. Reach out or follow-up in the comments with more resources to share with parents and teachers.

Friday, May 17, 2019

Future of Education

The future of education is present on the internet

I am not talking about MOOCs or SPOCs.

It lives on Youtube and Reddit servers, financed by their ad revenue. 

For example, if I want to learn Data Science I could enroll in some online courses... 

Or... I can dive in and start doing the work to develop the skills I need immediately. By watching youtubers and commenting on insider news I become a peripheral participant in the community of practice where I can freely test real algorithms on real data. When I start building a project I have interest in I can ask questions, share my challenges and accomplishments in chat communities on slack, discord, or twitter. To build clout in the python community, I can make some commits; or the R community, other commits. I can read open source code and follow leaders like these two from #Rstats Hadley Whickam and  David Robinson. The more serious I get, the more resources I discover.

Why pay for an expert's subjective structuring of content when you can structure your own learning based on your own personal interest and passion projects for free?

@E_Rushton #iteachmath  #mathchat #datascience

If you are unfamiliar with the youtuber 3b1b (Grant Sanderson) and/or his recent pi day video screenshared above. Experience it.

Monday, April 29, 2019

Emergent Curricula

As you may know, I was a cohort teacher for a 4-year NSF grant with the Algebra Project in 2009-2013. We had 2 years of curriculum developed ahead of time (not a textbook, per say) and 2 years of fly by the seat of your pants, here are some texts and full-length curricula to pull from, and hopefully we did enough during the 2-week summer workshop to prepare you - good luck!

Luckily, we weren't alone, and along with our university professor partners we would meet on weekends, co-teach some classes, and figure out how to structure material to meet the needs of our students.

This tweet by Anna Blinstein inspired this post.

Two areas of our class structure garnered follow-up questions from Anna. Just-in-time handouts and portfolios.

Just-in-time Handouts

All handouts were numbered, 3-hole punched, and expected to be kept in a 3-ring binder. I had the luxury of teaching in a sewing classroom with ample closet space for all students to keep their materials. Extra copies of all handouts were kept in folders in crates for absentees and lost papers.

In year 4, our Algebra Project cohort took discrete math and statistics as a double block. Here is a handout list, homework list, and journal list for my discrete/stats, algebra, and tutorial lab classes.

Here is a document containing the Spring Statistics Handouts. Notice that handout 12 split the class into 3 groups and differentiated the instructions of the performance task. We developed this task because our seniors didn't have a firm concept of percentage, and Bayesian statistics is lost without that.

You'll notice things like signature boxes asking students to commit to completing homework, because we were having homework completion issues (who doesn't?) and personalized messages to students and classes that you won't find in a traditional textbook.

Portfolio Tasks

Portfolio tasks asked students to find a mistake and correct it. I aimed to do this monthly, but in reality did 3-4 per semester. Here are some examples. Along with student portfolio submissions was a notebook and journal check - here are examples of the monthly-check format.


As a math coach I know that what works for one teacher in one context may have the opposite effects for a different teacher in the same or a different context. However, there are some universal points about classroom structure that are good for instruction.
  1. Have students define the major takeaways from lessons and activities.
  2. Have students recall previous major takeaways to apply to new lessons and activities.
  3. Have students refine major takeaways as new lessons and activities deepen their understanding.

kop _crush

Friday, March 15, 2019

Measuring Tape Radians Activity #piday

1) Give each group a 5-foot measuring tape roll.

2) Have them roll it around 4-5 fingers. The roll should measure more than 7 inches, otherwise re-wrap it larger. Then tighten it down to a little more than 6.25 inches (estimate 6.28").

3) How big is the radius of this rolled measuring tape?

4) Take predictions.

5) Measure from the outside toward the center of the roll.

6) Have students write reflections about how many radii fit around the circumference. Will this amount change for different sized circles?

Let me know how it goes in the comments!

Thursday, September 13, 2018

CMP@CPP: Task Design

I recently presented for the California Mathematics Project at Cal Poly Pomona and am sharing my talk and reflections here in hopes that someone can benefit and/or share their own thoughts and resources with me.

Stated goals

    Build trust
    Do math together
    Share strategies and resources

Here is a trimmed down video of the session along with the slides:

Overall Reflection

I felt more trust by the end of my session and feel that I was able to get many voices heard and shared during my talk. The introductory math task felt like a positive experience for everyone. Teachers shared their strategies and approaches in their small groups as well as during whole group discussions. I would like to get teachers to really visualize themselves in their actual classrooms, and plan lesson ideas around activities they intend to use with their students. This has proven difficult to do. Suggestions toward this goal are coveted.

Moment to celebrate

I was happy that I remembered to discuss the idea of taking estimates (albeit after the fact). I am not sure if making this mistake of ordering helped or hurt uptake to the idea of taking estimates before attempting a problem. I was proud that I was able to pull out of a single participants solution strategy to ask for another approach.

Missed opportunity

I intended to have multiple groups share-out after a period of small group work, and only had one group share. Then I shared my own approach. I think I made this adjustment because folks hadn't really gone down the path I intended. In retrospect, I should have shared a summary of my own approach before the group work so that the teachers had a clearer idea of what was expected of them. The conversations were too general and un-focused. Many teachers didn't talk about their lesson or unit planning at all, and the time felt less productive than it could have been. I also think that my circulating, listening, and probing was not where I want it to be in terms of active listening and pulling the nuggets from multiple groups in a short time.

What surprised me?

I was surprised at how quickly we used up the time and how little of my talk we got through. I plan to break it up into a few parts and try to deliver a more focused session on open questions later this year. My goals are the same, but my main evidence for success will be teachers leaving the session with problems they actually use in their classrooms, that have a fresh new feel thanks to our session.

Monday, August 6, 2018

Do the Math

What do folks mean when they say, "do the math"?

How about, "I'm not a math person"?

For what, by and large, do folks (randomly sampled from the entire population of English speakers who know the word
math) mean by math?

In a recent survey of LAUSD elementary teachers, most agreed with the statement "Some students have a natural talent for mathematics and others do not." There is a camp of math educators who disagree with this statement, but it seems they are in the minority. Is it possible that the primary confound in this conundrum is differing definitions of math?

To me, doing math means much more than the arithmetic that comes once a solution strategy has been followed to its natural end. Once a complex problem becomes a simple computation, the math is done, and the computers have homework.

I recently presented for the California Mathematics Project at Cal Poly Pomona and had this to say,

What are some things that the computer can't do that I am going to continuously do with my students to get them to reimagine what math is?
Over the course of two hours, I tried to make the point to a group of primary teachers that math is not about getting correct answers, rather it is about discovering patterns, the structure of solution strategies and critiquing each others reasoning.

Watching the entire video and writing this post, I realized some unstated goals:

    Demonstrate how math is a creative process
    Create dissonance around what it means to "Do the math"
    Create dissonance around what it means to be or not to be a "Math person"

If you or someone you know has a similar goal for their students, parents, or teachers; what are some strategies or suggestions you have to create such experiences for folk? I'd love to discuss ways to measure progress toward these goals in the comments.

This post is mainly a recapitulation of Lockhart's Lament and a resounding "hear hear" to Conrad Wolfram's 2010 TED, Teaching Kids Real Math.

SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.” We are losing so many potentially gifted mathematicians— creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle.

Yet even though we have read and watched these ideas, think to yourself the next time you say or hear someone use the phrase "do the math" or "I'm not a math person": has anything changed... yet?

Written for Sam Shah's Virtual Conference of Mathematical Flavors