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
  • - 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?

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!