Friday, December 11, 2015

Math Test Answers

Last week I posed 12 questions related to teaching elementary school math. The test was specifically designed for those who believe a traditional, "back to basics" approach to math is superior to the type of math we are trying to teach through the Common Core in 2015.

So here are the test questions and answers with the  first answer, the traditional math answer (TMA) and second answer, the 2015 Common Core (CCSSM) answer.

1. What is counting?
TMA - "one, two, three four five etc" with no sense of what the numbers actually mean.
CCSSM -  "one, two, three, four, five etc" understanding that three is one more than two and four is one more than three and that when you count five objects the word "five" refers to all five objects not just the last one counted. This is the idea of cardinality and is key to understanding number,

2. What is addition?
       TMA:  34
                + 48
       CCSSM The above is only an algorithm, a piece of arithmetic. Knowing just this is useless in
       problem solving. To problem solve you need to know the concepts of joining, separating,
       comparing and part-part-whole if you want to use addition or subtraction in anything useful. .
3. What is multiplication?
       TMA  112
                x    5
       CCSSM Just like addition and subtraction the above is just an algorithm, a piece of procedural   
       knowledge. It doesn't help you decide which operation to use in problem solving. You'll need to
      know the repeated addition, multiplicative comparison concepts as well as a few more to be know
      whether to divide or multiply
4. What is division?
       TMA:  5/115
       CCSSM See above
5. What is Pi?
        TMA: 3.14 etc
        CCSSM Yes, but this doesn't help a whole lot. Pi is a ratio between the diameter and   
        circumference of a circle. The circumference is always just over 3 times the diameter; or, if the
        diameter is 7 the circumference is 22.
6. What does the 0 mean in 308?
       TMA: A place holder
        CSSMIt means there are 0 tens. Place holder is a meaningless term than has no conceptual  
         value. Children need to learn initially that each place in place value has a value.
7. What is 1/2 divided by 1/4?
        TMA:   "Change the sign and invert the second fraction"
        CCSSM  This is virtual nonsense and doesn't help at all. Knowing that the problem is asking   
        how many 1/4s there are in 1/2 is of far more value especially in a world of calculators.
8. Make up a word problem for # 7 above.
        TMA: I can't
        CCSSM: How many 1/4s in the first half of a football game 
9. What do  you get when you reduce 4/8?
        TMA: 1/2  Unfortunately, most children being taught to "reduce" a fraction  think that 1/2
 smaller than 4/8 because the word "reduce" means to make smaller and the two numbers.1
         and 2 are  smaller than 4 and 8. I joke about reducing 4/8 to 4/8    
        CCSSM; You don't use the word  "reduce" fractions because children think 1/2 is smaller than
         4/8. You  rename or regroup 4/8.

10. Why do you  put a comma after every three digits in a large number?
         TMA: Because that's what you do
          CCSSM Because it marks the repetition of  100, 10, 1 in each place value referent 
11. What is area?
          TMA: length times width
          CCSSM this only allows you to measure the area of regular shaped objects. Area is the two
         dimensional concept of coverage. You can find the area of your hand by tracing it onto squared
         paper and counting the squares.
12. What does a degree measure in geometry?
         TMA: It measures angles
         CCSSM It measures the rotation of an angle about a point. This is the only way you can explain
         why an angle is 45 degrees 4 inches from its origin and still 45 degrees 4 feet from it;s origin.

Children must understand what they learn at the developmentally appropriate level; this is why the common core is so good.

Tuesday, December 8, 2015

Maths; The Way It Should Be

Last night marked the last class of probably the best group of graduate students I have ever had the pleasure of working with. The course, Math and Diversity, focused on teaching maths to children with diverse needs including ELL, Special education, poverty, maths disabilities such as dyscalculia, and working with mathematically talented students. The course focused on developing the students' relational understanding of math as well as a growth mindset. Here's a sample journal from Bria, one of the students in the class that is pretty typical of how all the students in the course saw their growth as math teachers this semester.

 "I am so glad that I took this class. My mathematics confidence has skyrocketed as a result of this semester. I used to be told that I was good at math, but I didn’t believe it because I was grouped with people who were notably good at math and did not compare to them. I was one of the weaker links in Mathletes, for example, and I felt that I struggled in my college calculus class. I know that I will never be as strong in math as many others, at least as long as it is not my academic or professional focus. But what I am learning is that I am better at mental calculations and quick consolidation of numerical information than many other people I come across in my everyday life. I was able to comprehend and work with new concepts presented in this class fairly quickly, and enjoyed being able to experiment with my new knowledge. By gaining conceptual understandings of things I had learned procedurally growing up, I am able to approach new math problems that come to me in life more thoughtfully, and I understand those thought processes. My new hobby is doing mental calculations of various operations and then analyzing exactly how I came to my answer. Feeling in control of math is a fairly new feeling, and I love it. It makes me more confident in my mathematics skills, and makes me more comfortable when I find myself out of control of math.
In the past I have not been able to be comfortable with accepting that something was challenging for me; I would admit defeat at the first sign of struggle. I understand that this is a common ailment of the person who has spent their childhood floating through academic requirements and being told that they are smart. But through the education I am receiving in the graduate program at Saint Michael’s, I am finally learning to practice what I preach. I am learning that finding something challenging does not put a blemish on my intelligence, and asking for help does not signify weakness or make the person asked think I am less intelligent than they are. I believe these things wholeheartedly when they come out of my mouth as a teacher, but I continue to struggle with it personally because I always prided myself on being “smart” growing up and worried that people would find me less so if I asked for help. But during the second half of this course, when a concept or problem presented in class was challenging I began to actually feel alright about admitting it and getting help from a neighbor, rather than chastising myself for not understanding something as quickly as my classmates. I think that the confidence I built from taking this class has allowed me to get over any math-related anxiety I used to feel, such that I now know I am “good at math.” I understand that people were not lying when they told me this in the past. With this newfound knowledge, I am comfortable with struggling. Finding math concepts that are difficult for me are not something to avoid, but something to tackle head-on. It’s fun now, and I have this class to thank for it."

Another student in the course felt that, as a child,  she was "a conceptual person trapped in a procedural world". I will miss teaching this course. 

Sunday, December 6, 2015

Maths and gender

So finally there is definitive research that shows the human brain of both the male and female of the species is the same. It comprises "a mosaic" of all kinds of characteristics regardless of gender. There is, therefore,  no biological  reason why boys should be better at math than girls.

This means that it is the context in which maths is learned that creates the inequalities, as Jo Boaler  so eloquently points out. It means that we, as teachers, as parents, as a culture need to adjust our expectations of what girls can do mathematically.We need to make sure that girls do not receive subtle or not-so-subtle messages that they cannot do math  and that the types of activities that occur in math class are not gender stereotyped in any way. 

Friday, December 4, 2015

A Maths Test

One of the wonders of the world is why math reform in the US has never seriously happened. It's as if traditional maths is part of the original US Constitution that cannot be changed. Recent commentaries such as this one in the NYTimes  and this one also in the NYTimes  begin to shed light on just what it is that causes this state of affairs. In spite of the overwhelming research evidence (e.g. Jo Boaler) that helping children understand math and not just memorize it is by far the best approach there are constant cries for "back to basics" and the traditional approach of memorizing facts and formulas.
So, for everyone who thinks this return to basics is the way to go here's a test to see how well you know the math that children  are expected to know in 2015.

1. What is counting?
2. What is addition?
3. What is multiplication?
4. What is division?
5. What is Pi?
6. What does the 0 mean in 308?
7. What is 1/2 divided by 1/4?
8. Make up a word problem for # 7 above.
9. What do  you get when you reduce 4/8?
10. Why do you  put a comma after every three digits in a large number?
11. What is area?
12. What is a degree measure in geometry?

Answers next week.