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7/28/2019 Submission on Understanding
1/2
I read and re-read the letter from members of the US National Commission on Mathematics
Instruction to Education Secretary Arne Duncan, which is foundhere.
I think the letter is excellent, and I hope it has an impact on Secretary Duncan and national
policy. However, we humans have a tendency to explain away and dismiss facts that are not
consistent with our beliefs, especially when we have strong feelings about the issue and arealready committed to action based on those beliefs. Chris Mooney has a good article on this
tendency called The Science of Why We Dont Believe Science. Given this tendency, can we
expect much, if any, response from Duncan.
I would say that, while this letter is not written in a hostile tone, it does take a frontal
attack approach to change, directly promoting principles that contradict current beliefs and
practices. Change would require a major rejection of current beliefs in which there has
already been much emotional, financial, and reputational investment. I think it is important
to argue the case made in this letter, but I would like to suggest that we can simultaneously
take another approacha Trojan Horse approach which does not oppose the current system
but seeks to improve it from within its current design, yet ends up undermining its poorpractices. This approach targets an area where we are ALL in agreement but in which the
current system is critically flawed. By concentrating on this flaw and making it obvious, we
force change.
Ive explained what I think the critical flaw is inthis guest poston Sue VanHattums blog.
(If you read my article, read the comments as well. I didnt discuss solutions to the Sarah
problem in my post because I thought it was important to leave the problem open. The extent
that readers have to work hard and struggle to find the solutions is the extent that my point is
made, because the math is very basic and we should easily see both conceptual solutions.
Theres closure to that example in the comments.)
In essence, the flaw is how we think about and measure understanding. Everyone wants our
children to understand concepts. No one is saying that computational fluency is sufficient
without conceptual understanding. But how we teach and evaluate teaching, and our use of
tests, is not always consistent with real understanding. When students compute the answer to
problems we assume they also understand the concepts involved in the solution. Too often
this is not a valid assumption.
We need to directly test for conceptual understanding instead of inferring it when students
can compute the correct answer, or worse, when they use test-taking tricks to select the
correct answer in multiple choice tests. For example, there is more than one way to use themultiple choice answers given with the problem to select the correct answer without actually
working through a solution.
An example of a multiple choice test that does assess conceptual understanding is theForce
Concept Inventory. A paragraph from the developers of the inventory:
https://mathematicsteachingcommunity.math.uga.edu/index.php/649/secretary-policies-support-teaching-profession-accountabilityhttps://mathematicsteachingcommunity.math.uga.edu/index.php/649/secretary-policies-support-teaching-profession-accountabilityhttps://mathematicsteachingcommunity.math.uga.edu/index.php/649/secretary-policies-support-teaching-profession-accountabilityhttp://m.motherjones.com/politics/2011/03/denial-science-chris-mooneyhttp://m.motherjones.com/politics/2011/03/denial-science-chris-mooneyhttp://mathmamawrites.blogspot.com/2013/06/guest-post-understanding-is.htmlhttp://mathmamawrites.blogspot.com/2013/06/guest-post-understanding-is.htmlhttp://mathmamawrites.blogspot.com/2013/06/guest-post-understanding-is.htmlhttp://modeling.asu.edu/R&E/FCI.PDFhttp://modeling.asu.edu/R&E/FCI.PDFhttp://modeling.asu.edu/R&E/FCI.PDFhttp://modeling.asu.edu/R&E/FCI.PDFhttp://modeling.asu.edu/R&E/FCI.PDFhttp://modeling.asu.edu/R&E/FCI.PDFhttp://mathmamawrites.blogspot.com/2013/06/guest-post-understanding-is.htmlhttp://m.motherjones.com/politics/2011/03/denial-science-chris-mooneyhttps://mathematicsteachingcommunity.math.uga.edu/index.php/649/secretary-policies-support-teaching-profession-accountability7/28/2019 Submission on Understanding
2/2
TheForce Concept Inventory is not "just another physics test." It assesses a students
overall grasp of the Newtonian concept of force. Without this concept the rest of
mechanics is useless, if not meaningless. It should therefore be disturbing rather than
comforting that students with only moderate scores on the Inventory may score well onconventional tests and get good grades in physics. Of course, experienced teachers have
learned to avoid problems that are "too hard" for the students. That includes mostqualitative problems that seem so simple until student answers are examined. Students dobetter on quantitative problems where the answer is a number obtained by substitution
into an appropriate equation, and even on harder problems that require some algebraic
manipulation. So should we not be satisfied that they have developed quantitative skills?After all, physics is a quantitative science! Or do we have here a selection process that
directs teachers to problems that students can answer with a minimum of understanding?
The Force Concept Inventory is a physics test. We need to develop mathconcept inventories.
From Wikipedia:
A concept inventory is acriterion-referenced testdesigned to evaluate whether a student has an accurate
workingknowledgeof a specific set of concepts. To ensure interpretability, it is common to have multiple
items that address a single idea. Typically, concept inventories are organized asmultiple-choice testsin
order to ensure that they are scored in a reproducible manner, a feature that also facilitates administration in
large classes. Unlike a typical, teacher-made multiple-choice test, questions and response choices on
concept inventories are the subject of extensive research. The aims of the research include ascertaining (a)
the range of what individuals think a particular question is asking and (b) the most common responses to the
questions. Concept inventories are evaluated to ensure testreliabilityandvalidity. In its final form, each
question includes one correct answer and several distractors. The distractors are incorrect answers that are
usually (but not always) based on students' commonly held misconceptions.
Developing math concept inventories is a large project, but worthwhile. I think that until we
make a clear distinction in our teaching and assessments between whatSkemp calls
Instrumental Understanding and Relational Understanding, we will continue to have our
students getting the right answers despite not understanding the relevant math concepts.
The ability of poor practices to nevertheless help students get right answers is what supports
those practices. To many people, getting the right answer is proof of understanding. Concept
inventories can dispel those people of this notion and reveal the inadequacy of poor
practices. Then those who believed in those practices will be ready for significant change.
Even individual teachers can include qualitative, conceptual questions in their own tests.
Doing so may help both teachers and students pay more attention to concepts. But what we
need in order to change the entire system is official, validmeasures of conceptual
understanding that expose poor practices for what they are, and shift the focus away from
simply computing the right answer to really understanding the material. The first step in
achieving this goal is to spread this idea as widely as possible, including through the US
National Commission on Mathematics Instruction.
http://en.wikipedia.org/wiki/Concept_inventoryhttp://en.wikipedia.org/wiki/Concept_inventoryhttp://en.wikipedia.org/wiki/Concept_inventoryhttp://en.wikipedia.org/wiki/Criterion-referenced_testhttp://en.wikipedia.org/wiki/Criterion-referenced_testhttp://en.wikipedia.org/wiki/Criterion-referenced_testhttp://en.wikipedia.org/wiki/Knowledgehttp://en.wikipedia.org/wiki/Knowledgehttp://en.wikipedia.org/wiki/Knowledgehttp://en.wikipedia.org/wiki/Multiple-choice_testhttp://en.wikipedia.org/wiki/Multiple-choice_testhttp://en.wikipedia.org/wiki/Multiple-choice_testhttp://en.wikipedia.org/wiki/Reliability_(statistics)http://en.wikipedia.org/wiki/Reliability_(statistics)http://en.wikipedia.org/wiki/Reliability_(statistics)http://en.wikipedia.org/wiki/Validity_(statistics)http://en.wikipedia.org/wiki/Validity_(statistics)http://en.wikipedia.org/wiki/Validity_(statistics)http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdfhttp://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdfhttp://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdfhttp://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdfhttp://en.wikipedia.org/wiki/Validity_(statistics)http://en.wikipedia.org/wiki/Reliability_(statistics)http://en.wikipedia.org/wiki/Multiple-choice_testhttp://en.wikipedia.org/wiki/Knowledgehttp://en.wikipedia.org/wiki/Criterion-referenced_testhttp://en.wikipedia.org/wiki/Concept_inventory