One of the things that I have enjoyed most about being a mathematics consultant at the Halifax Regional School Board is the outstanding group of educators that I get to collaborate with. My background is in senior high mathematics while most of the people I work with have a background in elementary and junior high mathematics education. I have learned tons about pedagogy, assessment and instruction as a result of professional conversations and collaboration with them.
Over the summer I had the chance to watch Tracy Zager's Twitter Math Camp (TMC16) Keynote talk. There were a number of things that really resonated with me but I especially took home the idea that it is important to learn from a variety of people. She stressed that teachers at different grade levels have a lot to offer each other and that all teachers would benefit from 'vertical collaboration'. Unfortunately for most teachers, this type of forum in which people from all different grade levels share and learn, is limited to online communities (such as the MTBoS and Global Math Department) and not in the brick and mortar world. I encourage you to check out Tracy's talk.
I was inspired by Tracy to share a few things that I've learned in the last year or so as a result of hanging out with elementary and junior high teachers. Many of them have to do with the importance of mathematical language.
1. Reading Three-digit Numerals
It turns out that when you read a three-digit numeral, you're not supposed to say 'and' between the hundreds and the tens digit. For example, the number 237 is read, "two hundred thirty-seven" and not "two hundred and thirty-seven." This is explicitly taught in grade 3 mathematics (3N02.01 Read a given three-digit numeral without using the word 'and'). When reading numbers, the word 'and' is reserved for the decimal. Using the word 'and' for the decimal is covered in grade 4 mathematics. For example the number 3.6 is read, "three and six tenth." I've heard it suggested that reserving the 'and' for the decimal might be a difference between American English and British English.
2. Reading Decimals
So to continue on with the 3.6 example, reading decimals is something that the elementary and junior high teachers that I work are very particular about. This is something that was also stressed during one of my math methods courses while working towards my B.Ed degree. The number 3.6 should be read, "three and six tenths" and not "3 point six" or "3 decimal six". I recently participated in a discussion on twitter with a group of teachers about this. Most teachers agreed that when introducing decimals to students, reading this as "three and six tenths" assists students with understanding place value. It also helps students as they move to write decimals as fractions. However, as students attain mastery of place value, there are situations where reading decimals with place value becomes cumbersome... how would you say 3.617829? (I've been told that if there were more than three decimal places you would say "point" or "decimal") Or how would you say $3.61 million dollars? I've always heard it read "three point six one million dollars" and not "three and sixty-one hundredths million dollars".
3. Reading Powers
The most recent thing I've learned about mathematics language is that the term 'power' is not to be used in the same sense as the term 'exponent'. In grade 9 mathematics, students are expected to identify the base, the exponent, and the power in an expression in exponential form (9N01). For example, the power 2^5 (where 2 is the base and 5 is the exponent), can be read “two to the exponent of five,” or “two to the fifth” and not “two to the power of five.”
4. 'Say More About That'
Reading 'Opening Minds - Using Language to Change Lives' by Peter H. Johnston made a powerful impact on the way I observe and participate in classroom discussions. Working with HRSB math coaches and support teachers has taught me how important not only what you say is but also how you say it. A phrase I have heard a number of coaches use is 'Say more about that". It lets the person that you are talking to know that they are respected and you're interested in what they have to say. It also lets you know more about what they are thinking.
5. Simplify a Fraction (added Oct. 27th, 206)
I learned today that, in the Nova Scotia curriculum, we simplify a fraction and use the terminology of simplest-term fractions. We don't 'reduce' a fraction to lowest terms because that makes it sounds like we are changing the fraction to make it smaller. The size of the fraction doesn't change, we are making an equivalent fraction that is in simplest-terms. (from Grade 5 N07.02 and N07.05). I was also recently reminded about using precise language when reading fractions. For example, the fraction 5/4 should be read 'five fourths' instead of '5 over 4' ( which can confuse students by focusing on the physical arrangement of the digits) or 'five out of four' (which doesn’t support understanding of a fraction as a single number).
I feel extremely fortunate to have a community of educators with a breadth and depth of different experiences with whom I can collaborate and learn at my workplace. What have you learned from the educators that you work with? Do you have the opportunity to have professional conversations with teachers from outside your school and/or grade level?
I recently visited Memory Lane Heritage Village in Lake Charlotte. It is a living history museum depicting coastal rural life in Nova Scotia during the 1940s. My five year old son had a great time. He especially enjoyed sitting in the 1928 Ford Model A car and pretending to drive. On the way back home I was thinking about other living history museums in Nova Scotia and realized that, based on my experience, it appears that the further away from my home in Halifax that I drive, the farther back in time the museums depict. When I got home, I dug up some data. Sherbrooke Village depicts a typical Nova Scotian village from the 1860s and the Fortress of Louisbourg allows you to experience life in Louisbourg during the 1740s.
I used Google Maps to find the driving distance from my house to each of these locations and discovered a nearly perfect linear relationship. How perfect you ask? The correlation coefficient was 0.99906. I quickly created a scatter plot with a line of best fit to show my wife. Despite my exuberance, she appeared to remain unimpressed.
It is also interesting to see that the points on the scatter plot are almost exactly where the sites are on a map of Nova Scotia as well. Mind Blown.
A question that I still have is whether this apparent temporal relationship is based on distance or displacement. Perhaps I need to collect some additional data (or not intentionally disregard data that doesn't fit my hypothesis)? If I travel in the opposite direction, should a living history museum depict life in the past or in the future? I'd love to visit Yarmouth some day to experience what life will be like in rural Nova Scotia in the year 2213!