It used to be that Math was lots of number crunching but Elementary Math has changed –my theory is the advent of the calculator and Google. Homework can no longer be a page of straight division problems as they can be Googled for the answer and even for all the steps involved!!!
I came across this working on homework with a 4th grader. Is it 9 divided by 2? or 9 divided by 4? Both have a remainder of 1.
See the diagram below for these two ways of interpreting division (albeit division without remainder).
Here is another example:
13 can also be grouped by quotition division see above with 3 groups of 4 with remainder 1. This method will yield the same Math results EXCEPT for problems like 12 divided by 5 which is 2 remainder 2. But divide 12 by 2 and we get 6 not 5 remainder 2 b/c the remainder goes in one more time (same goes for divide 21 by 6 is 3 R3 but 21 divided by 3 is exactly 7 not 6 R3.). So be aware that division these days is about interpretation and not computation.
Q: Tree A and tree B both have some birds. The tree A birds tell the tree B birds “if one of you comes to our tree, our population will be the double of yours.” The tree B birds tell the tree A birds “if one of you comes here, then our population will be equal to that of yours.” How many birds in each tree?
A: Tree A has 7 birds and tree B has 5 birds
This can be done with guess and check or algebra
This is for statement 1) A + 1 = 2(B – 1)
This is for statement 2) A – 1 = B + 1
This is A in terms of B from 2) A = B + 2
Then sub into statement 1 and solve for B
B + 2 + 1 = 2B – 2
B + 3 = 2B – 2
B = 5
then can solve for A using either equation A – 1 = 5 + 1 therefore A = 7
Yikes!!! She saw that superscripted exponent as just a plain vanilla x. How often do students not answer the exact question but something super wicked close? Scores do not always reflect student knowledge and skills due to these “fuzzy errors”.
People can improve their Math skills and scores by paying attention to the details to reduce fuzzy errors. I learned this the hard way in engineering school: a little m means milli or thousandth .001 buta big M means mega or million so if you write a M when you mean m your answer is off by a billionfold — that’s a BIG fuzzy error!
For lesson planning, multiple choice with “good wrong answers” can be used for instruction so that students see common errors and misunderstandings and get less fuzzy. Quantitative Comparison questions are amazing for this — they were sadly deleted from the SAT in 2005 but still alive on the GRE QC questions.
During exams, I walk around with a crayon or highlighter and make visual comments on student papers to point out details such as a missed negative sign (I obviously missed the above during the College Algebra final!!). This helps students to improve their metacognition and attention to detail which strengthens student skills, scores, confidence and enjoyment of Math.