The Right Answer is Not the Thing

This is not another post on assessment, though the content is related.  The central theme in this post is faculty being wise about the process of helping students navigate through mathematics in an efficient manner (something we might call “learning mathematics”) 🙂 .

As context, I want to share part of a lesson from our math literacy course.  Like many such courses, we both use accessible situations and recognizing patterns in the learning process.  This particular lesson uses interest (simple and compound) with these basic steps in the process.

1. We deposit $500 in an account that pays 6% interest each year, on that $500.  Find the interest earned in the first 4 years by competing the table.  [The table shows a row for each year.]
   Find the total money by adding the interest and the original $500.
2. We deposit $500 in an account that pays 6% interest each year, on the current balance (including prior interest).  Find the interest and current balance for the first 4 years by completing the table.  What is the total money for this account?
3. Which account results in “more money” for us?
4. We found the current balance by calculating “0.06 × 500 + 500”.  Is there a way to simplify this calculation so there is only one multiplication and no addition?

Of course, much time is used in the first two steps.  Students often have misunderstandings about percents, but these are motivational questions … as is #3.  However, the learning in the problem is all about the fourth step, which is looking for “1.06 × 500”.

Many teachers will present the 4th question in a manner that defeats the purpose of the question … “we added 6% to 100%; what do we get?”  This approach ‘works’ in that many students will see how we got the 1.06, and we feel good that they got the right answer.  Unfortunately, we just avoided all of the meaningful learning in this context.

First of all, students need to really know that percents do not have any meaning by themselves.  When we say “added 6% to 100%”, we have reinforced the basic misunderstanding that percents work like decimals in all situations.  It’s easy to determine if students have this misunderstanding by asking a variation of the classic question:

We had a 10% decrease in pay last year, and this year we got a 15% increase in pay.  Our current pay is what percent increase or decrease compared to the pay before the decrease?

This problem is tough for students because it does not explicitly state the core situation … that the base for each percent is the current pay … and we might think that this is the main reason we get the wrong answer “5% increase”. However, even when this fact about the base is pointed out, students continue to add the percents.

Secondly, the “we added 6% to 100%; what do we get?” question divorces the situation from the algebraic reasoning.  We’ve done adding of fractions, where a common base is required.  Somehow, with percents, we are comfortable leaving the base out of the problem when this produces more ‘right answers’.  Each of those percents has a base, which happens to be the same number in this ‘interest’ situation.  A more appropriate instructional move is to provide a little scaffolding:

Let’s write 0.06 × 500 + 500 this way:  (0.06 × 500) + (1.00 × 500)

Remember how we added 4x + 2x?  We got 6x.

Does that suggest how we might do the adding first?

Now, this instructional move will not make the problem easy.  The goal with this move is to connect the new problem to something fundamental in mathematics:  “like” things can sometimes be added.  Having the right answer without applying this concept is not learning any mathematics.

In our Math Lit course, this lesson introduces the concept of ‘growth factor’ which is then used as we identify sequences that are linear versus exponential.  That discrimination in sequences can get quite sophisticated, though we generally keep the level reasonable for the needs of the course and students.  The phrase ‘growth factor’ is used temporarily until we consider declining situations … however, this “adding to get one multiplication” is behind all exponential models.

Unrelated to the main point of this post, it’s interesting that many of us think of the number ‘e’ when exponential models are being discussed.  There are, of course, very good reasons why that is the most commonly used base in mathematics; unfortunately for the learning process, using base ‘e’ presents a disguise of the direct process involved in the situation … a multiplicative factor based on a percent increase or decrease.  I don’t see using ‘e’ prior to a pre-calculus course, in terms of helping students.

Back to the main point … whether you are teaching Math Literacy, Algebraic Literacy, or even the old-fashioned courses, “right answers” are a poor measure of the quality of learning.  The learning process itself needs to be richer and more valid than using a measure known to have limited validity.

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Assessment: Is this “what is wrong” with math education?

I have been thinking about a problem.  This problem is seen in too many of our students … after passing a test (or a course) the proficiency level is still low and understanding fragile.  Even accepting the fact that not all students achieve high levels of learning, the results are disappointing to us and sometimes tragic for students.

Few concepts are more basic to mathematics than ‘order of operations’, so we “cover” this topic in all developmental math classes … just like it’s covered in most K-12 math classes.  In spite of this, college students fail items such as:

• Simplify    12 – 9(2 – 7) ÷(5)(3)
• Write  3x²y without exponents

I could blame these difficulties on the inaccurate crutch called “PEMDAS”, and it’s true that somebody’s aunt sally is causing problems.  I might explore that angle (again).

However, I think the basic fault is fundamental to math education at all levels.  This fault deals with the purpose of assessment.  Our courses are driven by outcomes and measurable objectives.  What does it mean to “correctly use exponential notation”?  Does such an outcome have an implication of “know when this does not apply?”  Or, are we only interested in completion of tasks following explicit directions with no need for analysis?

Some of my colleagues consider the order of operations question above to be ‘tricky’, due to the parentheses showing a product.  Some of my colleagues also do not like multiple choice questions.  However, I think we often miss the greatest opportunities in our math classes.

Students completing a math course successfully should have fundamentally different capabilities than they had at the start.

In other words, if all we do is add a bunch of specific skills, we have failed.  Students completing mathematics are going to be asked to either apply that knowledge to novel situations OR show conceptual reasoning.  [This will happen in further college courses and/or on most jobs above minimum wage.]  The vast majority of mathematical needs are not just procedural, rather involve deeper understanding and reasoning.

Our assessments often do not reach for any discrimination among levels of knowledge.  We have a series of problems on ‘solving’ equations … all of which can be solved with the same basic three moves (often in the same order).  Do we ask students ‘which of these problems can be solved by the addition or multiplication properties of equality?’  Do we ask students to ‘write an equation that can not be solved just by adding, subtracting, multiplying or dividing?’

For order of operations, we miss opportunities by not asking:

Identify at least two DIFFERENT ways to do this problem that will all result in the same (correct) answer.

When I teach beginning algebra, the first important thing I say is this:

Algebra is all about meaning and choices.

If all students can do is follow directions, we should not be surprised when their learning is weak or when they struggle in the next course.  When our courses are primarily densely packed sequences of topics requiring a rush to finish, students gain little of value … those procedures they ‘learn’ [sic] during the course have little to no staying power, and are not generally important anyway.

The solution to these problems is a basic change in assessment practices.  Analysis and communication, at a level appropriate for the course outcomes, should be a significant part of assessment.  My own assessments are not good enough yet for the courses I am generally teaching; the ‘rush to complete’ is a challenge.

Which is better:  100 objectives learned at a rote level OR 60 objectives learned at some level of analysis?

This is a big challenge.  The Common Core math standards describe a K-12 experience that will always be a rush to complete; the best performing students will be fine (as always) … others, not so much.  Our college courses (especially developmental) are so focused on ‘procedural’ topics that we generally fail to assess properly.  We often avoid strong types of assessment items (such as well-crafted multiple-choice items, or matching) with the false belief that correct steps show understanding.

We need conversations about which capabilities are most important for course levels, followed by a re-focusing of the courses with deep assessment.  The New Life courses (Math Literacy, Algebraic Literacy) were developed with these ideas in mind … so they might form a good starting point at the developmental level.  The risk with these courses is that we might not emphasize procedures enough; we need both understanding and procedures as core goals of any math class.

Students should be fundamentally different at the end of the course.

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Active Learning Methods in Developmental Mathematics

We’ve all had this … students who attempt to complete a math course based on memorization; such students often report frustration when asked to apply knowledge in any way that differs from what they ‘learned’.  As mathematicians, we see the process of that frustration as a key part of what mathematics is all about.

As teachers, we sometimes behave in the same memorizing way.  For example, we attend a workshop session on a particular active learning method or methods.  After some planning, we begin to use those methods in our classes and usually feel good about the experience.

Any teaching method will be more effective if the teacher understands the whole story — how the method works and WHY it works, with connections to other knowledge about the learning process.

If you look for data and research on active learning methods, the results look very good.  However, most of that data collection is done by experts using the methods.  An observational study was done using a random sample of college biology teachers, with an eye towards seeing this positive impact of active learning methods.  Some faculty used little ‘active learning’, some used a lot, and some were in-between.  The results?  Well, not so good for ‘active learning’.  See http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3228657/?report=classic for the Andrews article “Active Learning Not Associated with Student Learning in a Random Sample of College Biology Courses”

It turns out that many faculty using active learning methods have memorized the ‘steps’ for the method but did not understand the method well enough to adjust it for their students … and also could not accurately monitor the process.  Putting students in pairs with a problem for ‘think-pair-share’ will not automatically produce better learning.  Understanding is important for us, as well.  The authors of the study listed above believed that was the critical factor for not seeing positive results.

I’ve been saying something for 20 years, and believe it is still true today:

Developmental mathematics … we are a desperate people!

Partially because we’ve been teaching the wrong courses, our work has not been successful over a long period of time and over a large range of locations.  That process results in us looking for something … almost anything … that might help in our classrooms.  In many ways, this is the same attitude that our students bring to our developmental math courses.  We see something — it might help, so we quickly try it out in our classes.  Learning a new teaching method is like any other learning: there is a process, and ‘knowing steps’ does not equal ‘learning’.

Here are some pointers on how to use new teaching methods effectively … ‘active learning’ or otherwise.

1. Experience the method yourself repeatedly:  for example, use the think-pair-share process to learn something new.  Look for the how & why of the method, and develop an intuition for what it looks like when it works.
2. Read and use multiple sources of information.  You are most likely hearing about a method from somebody who heard about it from somebody who heard about it … each of those stages involves filtering and distortion (just because it’s human communication).  Multiple sources will provide a more accurate picture of the method.
3. Use the engineering principle: estimate the time it will take, then double the number and use the next larger size.  “10 minutes” becomes “20 hours”.  That’s a little extreme, but valuable as a guideline … nothing breaks a teaching method quicker than rushing it.  This applies to both your planning time, and to the operational time in the classroom for the method.
4. Don’t be deceived by appearances and initial student reactions, which are often skewed (more positive) by ‘something new’.  Assess the results using multiple measures — direct observation, one-minute paper, survey, quiz, etc.
5. Assume that your first use is a crude approximation requiring a number of adjustments based on analysis of results.  Proficiency is the result of lots of practice … and learning from that practice.
6. Allow yourself to reject one method and switch to something else.  We all need to become effective teachers, but we don’t need to become the SAME teacher.  Use methods you can be enthusiastic about, since that helps students almost as much as the details of the method.
7. Talk about your experiences with colleagues you trust.  You are learning something new, and it’s complicated … verbalizing helps your brain clarify the process and the results.  Ideally, you would form a ‘lesson study’ type group working on the teaching method.

I wrote those pointers with active learning methods in mind, but they apply to any method — including lecture (aka “direct instruction”).  Lecture, sometimes defined as continuous elaboration by the ‘teacher’, is a valuable tool for us; it’s just not adequate in general, and needs to be used intentionally.  My own classes tend to be several lectures of 5 to 10 minutes separated by some active learning method.  You might experience ‘experts’ who claim that we don’t remember what we hear; the irony is that the message the expert is delivering is one which they HEARD.   The critical thing is to have the learner’s brain engaged with the material using multiple teaching methods appropriate to the content; listening is an effective method for some things.

We each need to develop high levels of skills with a variety of teaching methods, because that is what experts do.  Limiting our methods, or using methods poorly, impedes our student’s learning … or even causes damage to their learning.  Just like our students, we need to have a growth mind set.

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Data on Co-requisite Statistics (‘mainstreaming’)

Should students who appear to need beginning algebra be placed directly in a college statistics course?  For some people, this is no longer a question — they have concluded that the answer is an unqualified ‘yes’.  A recent research study appears to provide evidence; however, the study measured properties outside of what they intended and does not answer a basic question.

So, the study is “Should Students Assessed as Needing Remedial Mathematics Take College-Level Quantitative Courses Instead? A Randomized Controlled Trial” by Logue et al.  You can read they report at http://epa.sagepub.com/content/early/2016/05/24/0162373716649056.full.pdf

The design is reasonably good.  About 2000 students who had been placed into beginning algebra at a CUNY community college were invited to participate in the experiment.  Of those who agreed (about 900), participants were randomly assigned in to one of 3 treatments:

1. Elementary Algebra regular    39% passed
2. Elementary Algebra with weekly workshops   45% passed
3. College Statistics with weekly workshops    56% passed

At these colleges, the typical pass rate for elementary algebra was 37% while statistics had a normal pass rate of 69%.

The first question about this study should be … Why is the normal pass rate in elementary algebra so appallingly low?  I suspect that the CUNY community colleges are not isolated in having such a low pass rate, but that does not change the fact that the rate is unacceptable.

The second question about the study should be … Would we expect a strong connection between completing remediation (or not) with performance in elementary statistics?   The authors of this study make the following statement:

it has been proposed that students can pass college-level statistics more easily than remedial algebra because the former is less abstract and ses everyday examples

In other words, statistics is not abstract … not mathematics at the college level.  The fact that statistics focuses on ‘real world’ data is not the problem; the fact that the study of statistics does not involve properties and relationships within a mathematical system IS a problem.  I’ve written on that previously (see “Plus Four: The Role of Statistics in Mathematics Education at https://www.devmathrevival.net/?p=976)

The study uses ‘mainstreaming’ in their descriptions of the statistics sections in their experiment; I find that an interesting and perhaps better phrase than ‘co-requisite’.  It’s unlikely that the policy makers will move to a different phrase.

The authors of this study conclude that many students who place into elementary algebra could take college-level math (represented by statistics in their study) with additional support.  The problem is that they never dealt with the connection question:  How much algebra does a student need to know in order to succeed in basic statistics?  The analysis I am aware of is “not much”; in the Statway (™) program, most of the remediation is in the domains of numeracy and proportional reasoning … very limited algebra.

This is the basic problem posed in all of the ‘research’ on co-requisite remediation:  students are placed into low-algebra courses (statistics, liberal arts math), and … when they generally succeed .. the proclamation is the ‘co-requisite remediation works!’.  That’s not what is happening at all.  Mostly what the research is ‘proving’ is that those particular college ‘math’ courses had an inappropriate prerequisite of algebra (beginning or intermediate).

Part of our responsibility is to explain to non-math experts what the relationships are between various math courses, using language and concepts that they can understand while preserving fidelity with our own work.  We need to make sure that policy makers understand that it is not an issue of us ‘not wanting to change’ … the issue is that we have a different understanding of the problem and potential solutions.  In many colleges, the math department is already ahead of where the policy makers want us to ‘go’.

I encourage you to read this study thoroughly;  Because it using a ‘control’ and ‘random assignment’ design, this study is likely to become a star for policy makers.  We need to understand the study and provide a better interpretation.

 
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