7/28/2019 Submission on Understanding

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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-accountability

7/28/2019 Submission on Understanding

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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