*This post is a reply to a thoughtful educator that I respect and with whom I disagree on certain education-related topics. The stir began with my tweet,


I replied to Shawn,

Dancing in a math lesson will not improve thinking. THINKING advances mathematical thinking.”

A cute “engaging” math song might energize the kids but it won’t make them better at math. Surely, “memory is the residue of thought” and it is actually the main key to thinking (see D.T. Willingham’s posts on cognition and learning), but to use dance in 2nd grade as a way to memorize subtraction facts is not the most effective way. Despite the general belief that testing is damaging, cognitive science demonstrated that testing has a far greater impact than additional study. So if you want kids to memorize their number facts (so as to give space to higher-order thinking in solving problems) instead of making them dance it is better to allow them to self-test or to test each other in pairs.

I did an experiment to test this cognitive science claim, and, coincidentally, it related to number facts memorization. Each student had a set of number facts flashcards (i.e. front – the addition 8+9, back – the answer 17) and paired up with another student. I had them test each other (showing just one side of the card) for 10 minutes on a daily basis.


  1. The progress within 4 weeks was mind-blowing. I kept records for 4 weeks in a row and wrote down the number of facts each kid could do correctly within the same amount of time): from an average of 20-25 facts in the first week all students could do OVER 45 at the end of the 4th week.
  2. The progress students made within the first 2 weeks allowed us to move to challenging mental math activities (i.e. from 8+9=17, they could progressively do “Double the 9, add 10 and subtract 17) really fast. I kept records of all this data so as to help me push their thinking further in problems and inquiries.
  3. Each 10-minute buddy practice was followed by whole-class discussions (we gathered on the carpet). Questions:

What did you find difficult?

Why are some numbers easier to add than others?

What strategies did you use?

Is that …. always true?

How is …(i.e. 9+ 3) similar to …(19 + 3)? How do you know? etc.

This allowed students to reflect, think and share.  (I talked about language and thinking here  ). Many learned new strategies through this act of sharing thinking (i.e. 19+3 can be done as rounding up to 20 and adding just 2 instead of 3).  I would often video record these shared sessions and upload the videos to the class blog so the students could revisit them. Shared thinking at the end of an activity also helps the teacher spot misconceptions.

  1. Testing each other was met with high enthusiasm by all students (despite my initial fears that it would bore them). As a matter of fact, when I had to skip the number facts practice a few times (either because we had a Skype session, a guest speaker was invited or whatever the reason) the students were “upset”.
  2. I preferred the buddy system although self-testing works, too. You can use it at times but I like students to collaborate if there are two equally effective alternatives to learning.


Practice is critical to empty working memory so as learners can devote their attention to really difficult tasks and reduce the cognitive load. That is not to assume that we don’t INQUIRE into numbers and operations. Inquiry leads learning in IB schools and it is central to understanding. Thus,

  • from explorations of math in the real life
  • to wondering – and remember, these are 2nd graders’ and also 2nd language learners’ questions (i.e. Are graphs related to measurement, data/numbers or both? – this actually stirred a lot of debate and we concluded that it relates to both because you cannot create a correct graph unless you have correct data AND display the information correctly – i.e. the bars in the bar graph have to be equal length-wise)
  • noticing (i.e. “Miss Cristina, it is 12:40 AND there is an obtuse angle on the clock!” – connecting geometry concepts with time measurement spontaneously)
  • questioning (i.e. How do you know? Is there another way we can prove…? Does this… always make sense- can we make it a rule? What if we changed… (a number, an operation etc.)?)
  • collaborating (i.e. building the most complex shape together and explaining it to others)
  • and thinking (I think, for instance, cracking the pattern in the Fibonnaci sequence in 2nd grade is really amazing)

we do a lot to gain understanding (read my entire series of posts on that).

*See some photos in our math class – we blend inquiry, practice (either through games or other types of activities),  and collaborative plus independent work. You can also notice that THINKING and QUESTIONS are made visible everywhere in the classroom (yes, I hate cute teacher-made posters or anchor charts).

Measurement questions 3 pizap.com13586778861071 pizap.com13780433528541 pizap.com13780436067101 pizap.com13780437355731


………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………. My advice:

1. Read widely everything that is essential to education. Blogs and Twitter exchanges are not enough – they are just snippets of learning. Education and all that it encompasses IS rocket science – think only how many dimensions we need to address from curriculum to student motivation.  If you *do* care about your students’ learning, read BOOKS on education, cognition, psychology, pedagogy. Only a well-read educator can truly think critically about his/her own practices, compare theories, and eventually become better.

2. Dismiss any theories that are not supported by evidence and extensive research. For instance, BrainGym has no research to back it up except 2 classes (yes, you read that correctly) of 7-8 year-old children. Two classes to be the basis of a learning theory? You must be kidding.

Learning styles, the myth of the learning pyramid (you know, the one that tells us we learn only 20% by reading…) and other brain-based pseudoscience claims have to cease to intoxicate the educational environment.

3. Engage in constructive debate. If you listen to and follow only educators that are like-minded you only reinforce your ideas – confirmation bias does not avoid anyone, educators included.  See what the “other” side have to say, whether you are a “progressive” or a “traditional” educator.