MLB 1 What is Math?

29 x 12 = The Big Question

I solved 29 x 12 = using the standard algorithm, following the steps I teach my students.  First, I estimated the answer by rounding the multiplicand and multiplier to the nearest ten: 30 x 10 = 300.  I know my answer will be higher than 300, by about 30.  Then, I set up the algorithm vertically and began multiplying the numbers in the units column.  I ended up with 29 x 12 = 348, which is appropriately close to my estimated answer.

For a fleeting moment, I thought about solving it in my head, by cutting up the numbers into 30 x12 and 9 x12, but that felt like a lot of brain power on a Friday night, when using the algorithm is quite efficient for me. 

The TERC video featuring 5th grade students explaining their solutions to the same problem demonstrated impressive logical thinking and articulate explanations.    The first student explained that he split his multiplicand into 20 and 9 and multiplied them separately by 12. He then added those two products together for his final answer. He described the 20 as 2-10s, and easier with which to work.   To check his work, he did repeated addition of 29, 12 times and found the same answer: 348.  One of his classmates noticed that his work paper showed additional work, involving a 90, and asked for clarification.  The student explained that he had also split the multiplier in two numbers, 10 and 2, before multiplying them by 9.

The second student shared a different approach to solving 29 x 12.  She used repeated addition as her beginning strategy.  However, she read that she rounded the 29 to 30 for use in the problem, because 30 was an easier number with which to work.   After adding 30, 12 times, her answer was 360.  She then subtracted 12 from t360, representing the 12 extra 1's that were added by using a rounded number, instead of the original 29. 360 - 12 = 348.  When the teacher asked for clarification about using 30s, instead of 29's, a classmate explained her process of substituting 30 for 29 and subtracting the "extra" 12 for accuracy.  

When watching the TERC video the second time,  I noticed the authentic understanding of the students that became clear when they were asked by their peers to clarify their thinking or restate a strategy used by a classmate.   When reflecting and explaining their mathematical thinking "off-script" one could see that the understanding was genuine.  The first student was able to use two strategies to solve the problem - the first illustrating conceptual understanding (splitting the numbers into smaller parts) and the second expressing a more efficient means of calculating the answer (checking with the standard algorithm)   Both the presenters and the listeners engaged fully in the activity.  

The role we saw the teacher perform was that of facilitator - disguised as curious on-looker - directing the students to explore and explain their thinking through exploratory questioning.  This role allowed the teacher an opportunity to see the depth of student understanding and engagement, as well as see misconceptions a student may have created.   Though we did not see any examples of this in the short video, such mathematical conversations among students and teachers allow both to see misconceptions and provide an opportunity for reframing ideas, or a-ha! moments, through the sharing and explaining process.   Assessing and teaching in one seamless activity (if the stars are aligned correctly, anyhow.)

Mathematics is the discovery and use of patterns to make sense of the world.  Mathematical thinking involves the process of looking for and recognizing these patterns and applying them to life.  That translates into a different kind of maths in the classroom - one where students are completely engaged in their thinking.   These engaged students create their own understanding of mathematical concepts based on experience and experimentation.  This requires patience and facilitation (and behavior management!) skills on the part of the teacher, and persistence and risk-taking on part of the students.