Opinion – CCM may be Unproven
One argument I’ve seen against CCM is that the standards are
unproven. Since they were made up by math teachers, I believe that they are (1)
correct and (2) well structured. Therefore, it is hard for me to see how
learning them would do any damage. Moreover, to allow the sharing of teaching
materials and to level expectations across state lines, they are being widely adopted
and implemented in the USA. Therefore, not learning them could put your
children at a disadvantage to their peers. Why would you choose to risk their
futures rather than help them master the standards?
My children graduated high school before CC Math was
introduced. However, if they were still in school, here is how I would approach
the above question. I would hedge my bets – on the off chance that the
standards would cause them problems, I would show my children the old-fashioned
“we’ve-always-done-it-this-way-and-it-was-good-enough-for-me” way – and help
them learn both. While comparing old and new, if I or one of my children did
discover a flaw in the CCM standards, we would be richer for the experience. Alongside
the standards, I would share with my children lessons learned regarding how to
“befriend” computers and programming languages. For example, the object
paradigm, fundamental to many modern programming languages, can help us
formulate problems such that computers can solve them effectively.
Anecdotal reports (available on-line) say that, taken
step-by-step, CCM standards are working well for many motivated teachers and
students. Other anecdotal reports say
that students are reduced to tears by new problems and unfamiliar approaches
(unfamiliar to their parents) inherent in both the CCM standards and the way
some teachers are using them. I will not
debate either set of reports, but it is reasonable to observe that any academic
discipline requires competent teachers and helpful teaching materials, and that
CCM is still relatively new.
While the standards may have been “vetted” by mathematicians
and math teachers, it is not intuitively obvious that all standards are
“age-appropriate.” The standards creators do write that, “the development of
the standards began with research-based learning progressions detailing what is
known today about how students’ mathematical knowledge, skill, and
understanding develop over time.” This shows an awareness of the importance of
this matter; however, there is not yet enough evidence to demonstrate that the
current standards are, in fact, age appropriate. That children mature
intellectually at different rates leaves open the question of whether the new
standards are right for all students at all times.
This is not a trivial matter, but if your child is one of
those for whom certain concepts are out of reach, I suggest patience. It is
unlikely that she is “stupid;” it is more likely that she is not quite ready
for the particular standard that is part of her homework that day. One issue
that is sometimes overlooked during the early primary grades is that students
in one class may differ in age by almost a year. Younger students may appear to
be slower learners when in fact they are simply younger and less mature than
their classmates. By the same token, if your child is older, be thankful but
remember, his age advantage will diminish over time.
Teachers’ skills matter, but as I said earlier, children in
public schools, private schools, and even home schools will be faced with
teachers of varying skill and motivation levels. As a parent, you want the best
possible learning environment for your children. However, I believe that you
will agree that it is not useful to assess the value of CCM based on the
competence of available teachers. Granted, an incompetent teacher increases the
challenge to student and parent alike to learn the material. However,
overcoming this impediment can be seen as a positive experience. In other words, while teachers’ skills
matter, student and parent attitudes matter more.
The claim of the education community is that, a
comprehensive understanding of CCM will be an advantage in college and in the
world of business. It is your right to
be skeptical, but if that claim turns out to be true, would you prefer that
your children have that advantage or that they compete with other (parents’)
children at a disadvantage? Over time, colleges, and universities as well as
many businesses are likely to expect that earning a high-school diploma will
imply mastery of CC Math standards. However, given the pace of technology
change, CCM may be necessary but not sufficient.
CCM Lesson of the day: CCM Practice Standard 6
Attend to precision
“Mathematically proficient students try to communicate
precisely to others. They try to use clear definitions in discussion with
others and in their own reasoning. They state the meaning of the symbols they
choose, including using the equal sign consistently and appropriately. They are
careful about specifying units of measure, and labeling axes to clarify the
correspondence with quantities in a problem. They calculate accurately and
efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give
carefully formulated explanations to each other. By the time they reach high school they have
learned to examine claims and make explicit use of definitions.”
My Comment
This step conflates accuracy and precision, terms which do
not mean the same thing. It also uses the word precision (underlined in the
paragraph above) to explain what precision means. The modal “attend to” is
vague enough that we can take this to mean ‘use lots of precision’ or ‘be
consistent in your use of precision.’
Process implications:
·
When solving a problem, estimate the solution
and compare your result for accuracy
·
When solving a problem, consider the precision
you will require.
Definitions
Accuracy and precision are
not the same thing. For example, in measuring the distance from your house to
the nearest post office, an accurate estimate (to the nearest mile) might
indicate how long it will take you to make the trip by car.
However, if you need to lay
out cones on a soccer field, you might need a more precise measurement, in feet
say, to know how many cones or how much gypsum you will need.
Accuracy: The closeness of a result to the “true”
value.
Precision: The degree to which your result must be
refined.
Applications and Examples
Accurate: If our gas gauge indicates half a tank, accuracy
could be critical so that we realize how far we can go, but it’s unlikely that
we can justify being much more precise than to estimate our remaining mileage
to within ten miles, as in “we should be able to travel another 160 miles.”
Precise: By contrast, when using a laser to perform a
medical procedure, precision in fractions of millimeters may be required to
perform effectively and safely.
Accurate and Precise: When preparing to install new
flooring material in a room in your house, your measurements must be accurate
and precise. You will want to measure accurately (the correct distances,
in feet and inches) so that you know how much material to buy. You will want to
refine your measurements precisely to 1/8 inch or less (differences
between the measurements of floor and flooring material) prior to cutting the
material in order to avoid unsightly gaps or bulges.
Opinion – CCM Politics
I am well aware that some parents have chosen to fight
against CCM standards. Protesting the
teaching of CCM does not seem helpful to me; however, I have not been involved
either in teaching public-school math or in formulating the standards. If you
have had these experiences and can show that CCM is an inferior approach, I
trust that your criticisms will be constructive, designed to help improve the
standards or the teaching of those standards. If you are not part of such an
insider group, blaming the standards on big government, a political party, or
unions strikes me as changing the subject and quite beside the point.
There is no agreement regarding who “won the battle” to make
CCM a national standard. Proponents and opponents can each claim partial
success. However, this begs the question of whether learning CCM benefits
students. Moreover, the opposition has only the most rudimentary idea of how to
replace it. In any event, even if the “I-hate-common-core” side knew exactly
how to replace it with something better, by the time they achieved their goal,
the affected children would have moved on.
Given that so many states have adopted (or adapted) CC Math
standards, I believe that, rather than resist, a better choice is to help our
children learn the standards. However, I will offer these observations:
1)
The transition from one pedagogical approach to another
is inherently difficult. If we learned to eat meat the good old American way, by stabilizing it with a fork using the left
hand and cutting it with a knife using the right hand (for right-handed
people), putting down the knife, picking up the fork with the right hand, using
the fork to transfer the cut piece from plate to mouth, and then transferring
the fork to the left hand to restart the process, we may find it awkward at
first to change to the more efficient European approach: i.e., simply holding
the knife in the right hand and the fork in the left throughout the cutting and
eating process.
2)
Learning the standards is not the same thing as
learning to solve problems. Therefore, I will continue to emphasize a process
for problem-solving.
3)
A growth mindset and practice will be required for you
and your children to learn new methods to perform familiar tasks in math. If
math were easy, we would not need to study it.
4)
One issue regarding the transition to CCM arose very
early on: since standards at one grade are built on those from the previous
grade, asking teachers and students to make the transition to CC Math standards
at all grades simultaneously was unrealistic. This should no longer be a
concern.
5)
A real concern is whether, having mastered CCM, a
student is prepared for the information age and the world of computers and
computer programs. I suggest that we cannot rely entirely on math education to
prepare students to engage productively in the modern world of technology,
especially since technology evolves faster than formal education. Rather, we
(as guides, parents, and teachers) must be alert to what students need and help
them to fill in gaps in their formal education (including CCM). This must be a
team effort.
CCM Lesson of the day: CCM Practice Standard 5
Use appropriate tools strategically.
“Mathematically proficient students consider the available
tools when solving a mathematical problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software.
Proficient students are sufficiently familiar with tools appropriate for their
grade or course to make sound decisions about when each of these tools might be
helpful, recognizing both the insight to be gained and their limitations. For
example, mathematically proficient high school students analyze graphs of functions
and solutions generated using a graphing calculator. They detect possible
errors by strategically using estimation and other mathematical knowledge. When
making mathematical models, they know that technology can enable them to
visualize the results of varying assumptions, explore consequences, and compare
predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as
digital content located on a website, and use them to pose or solve problems.
They are able to use technological tools to explore and deepen their
understanding of concepts.”
My Comment
This step suggests that rulers, protractors, compasses, calculators,
computer programs, spreadsheets, programming languages, etc., are useful when
they are useful. It also suggests that knowing how and when to use these tools
builds problem-solving capability.
Process implications:
·
Learn to use tools, particularly computer
programs such as spreadsheets.
·
Use tools to verify problem solutions generated
in other ways.
Definitions
Digital content: refers to information stored in a
computer system.
Applications and Examples
Use a spreadsheet to record a baseball team’s batting
records and compute averages.
Use a spreadsheet to construct a vacation plan including
1.
Things to assemble and pack for the trip
2.
Tasks for each family member and when they must be done
3.
A travel schedule showing dates, destinations,
distances, and means of travel
4.
Math activities to be performed during the trip 😊
