Discovery Learning in Developmental Math

Compare these two situations:

You are placed in a laboratory with 12 objects of varying shapes and matching openings on the wall.  The learning goal is for you to discover which opening goes with which object.

You are placed in a room with 3 sheets of paper, 3 pencils, and two other people.  The learning goal is for you to discover the properties of a good drawing of a rose.

Really, take a minute to see yourself in each situation.

When we design a class to depend on discovery learning, we often assume that the students will experience a good thing.  We’ve listened to the sages say ‘guide on the side’ (rhyming makes right!!), so we have stepped out of the way.  If students discover the math, they will own it and learn it better (or so the story goes).

Like other people doing reform courses, I have been using discovery learning more.  From that point of view, the most important thing to say is this:

Discovery learning is very difficult to design and implement for positive results.

Two problems routinely come up with discovery.  First, many students have a difficult time seeing the idea that we want them to discover; this is primarily a communication issue.  The second: students come with prior knowledge, some not so good; the ‘discovery’ process often activates erroneous patterns, and the student ‘discovers’ initially that they had a great thing — which then needs even more effort to re-direct to better ideas.

The instructional materials being developed for the emerging models (Math Lit, Carnegie Pathways, Dana Center Mathways, commercial texts) tend to build a discovery process in every lesson, centered around a sequence of questions.  In general, the materials insert ‘check points’ for the instructor to assess the quality of the learning.  The developers work hard to create these materials that can be used by a variety of faculty.

The research base for discovery learning has not been consistently positive, and I think we sometimes confuse the motivational impact with learning mathematics.  I did some searching for a productive analysis of the issues with discovery learning, and found something that might help us.  The article is by Kirschner, Sweller, and Clark; called “Why Unguided Learning Does not Work …”, and is available at http://www.ydae.purdue.edu/lct/hbcu/documents/Analysis_of_the_Failure_of_Discovery_PBL__Experiential_Inquiry_Learning.pdf

This article is not a quick read.  However, I think you will find useful information which will impact your work in the classroom.

Join Dev Math Revival on Facebook:

Contextualized: Does Everything Need an Application?

In some corners, ‘algebra’ is getting a bad reputation.  Algebra weeds students out of programs, prevents completion, and is not identified as needed for most jobs.   Some of us have responded by taking a very  contextualized approach to algebra, so students can see how useful it is.

This is the first week of our summer classes, so I have been working with my introductory algebra class on basic concepts.  We actually do very little with operations on signed numbers (traditionally the start of an algebra course); instead, we spend 3 class days mostly working on the language of algebra.

My interest is in having each student understand the objects we are using.  When we see ‘3x’, I want them to have multiple and correct ways of expression this verbally.  When we see ‘the square of a number’, I want them to have at least one correct symbolic expression they can write.  I deliberately do all of this work without any context for each problem; in other words, the problems are not framed in terms of a situation with physical objects or meaning.

In our Math Lit course, we also do some of this same work.  The difference there is that we introduce algebraic reasoning by talking about some contexts where algebra might be helpful, and then deal with understanding the objects when there is no context.  Does it help to have the context first?  Not really.  It’s fun to have a context, and it motivates some students (though not most).

What seems to happen with context is that ‘understanding the context’ takes quite a bit of energy; I think the brain tends to then organize related information as being connected to that context.  Making the ‘math visible and general’ is not easy, when students begin in a context.  In some ways, beginning in a context comes across as just being a more complicated puzzle word problem (“two trains left at the same time …”).  Students seem to feel like the context was just there to give them another word problem.

One of the myths seems to be that “we need to make it relevant”.  In some cases, we have gone so extreme that we refuse to cover a topic if we can not show students a context that they can see the math within.  I think we have confused math education with something else — having a context for everything is a basic property of occupational training.  Unless we are teaching an occupational math class, context is a tool to use when it helps; context should not be a cage that prevents good mathematics from being learned.

Whatever we might call a course (introductory algebra, mathematical literacy, whatever), a core understanding of basic ideas is critical.  Think about this problem:

2x+4x=??

Without further learning, something like 30% of students will give either 6x² or 8x² as an answer.  [Even among those who generally give the correct answer, their confidence may not withstand a little questioning about ‘why’.]  I’m not talking here about understanding operations on rational expressions, or factoring trinomials with a leading coefficient greater than 1, nor about simplifying radicals with an index of 3 and a radicand containing constants and variables.  The issues here deal with the initial constructs of an algebraic language system.

A related issue is ‘transfer of learning’ — context generally creates barriers to transfer.  Context is a concrete approach, and serves an instructional purpose when used appropriately.  However, an initial learning (in context or not) does not enable transfer to situations where the knowledge is needed.

In reforming the math curriculum, we need to keep aspects of the prior design that have benefits for students.  Think about (1) Transfer of learning and (2) Student confidence.  Known factors support transfer of learning — ease of recall, connections, and flexibility.  Student confidence seems to be impacted by feedback and repetition.  The presence of repetition can support both transfer and confidence — it’s not the presence of any repetition; rather, it’s purposeful repetition (including the use of mixed repetition) that provides the benefits.

When people say that algebra is not needed in occupations, this is often based on people in those occupations looking at a list of typical topics in an algebra course.  I think different results would be obtained if we asked about a different list — variables, algebraic reasoning, functions and models, graphical interpretation, etc.

I’d encourage us, as we re-build our curriculum, to incorporate more context — but not be limited by context.  I’d encourage us to help students learn deeply by providing sufficient repetition (with mixed practice especially).

 Join Dev Math Revival on Facebook:

Mathematical Literacy and Equity

I just finished watching a talk given by my friend Uri Treisman at the NCTM conference, in which Uri presents some great sets of data and a wise viewpoint on the theme of equity.  Seemingly unrelated, I am often asked “what is so different about that Math Lit course?”

Most of the data I have seen suggests that the traditional developmental math curriculum tends to reinforce existing achievement gaps.  Students who had done well overall, but not in math, pass our courses.  Students who have struggled pass at a much lower rate.  Access is not the same as equity.  In particular, minority students tend to have much lower pass rates than majority Caucasian students.

In Uri’s talk, he tells the story of how Boeing became successful at building airplanes … by designing ‘fault-proof’ planes, where one failure would not cause a catastrophic event.  Uri calls us to design fault-proof educational systems to avoid catastrophic events for our students.

A Math Lit class is one attempt at a fault-proof course.   In the traditional curriculum, there is a tendency for students to be defeated by mathematical ideas that they did not understand.  The Math Lit approach for this problem is to avoid catastrophic failure; within each class, we identify students who did not understand enough to succeed and provide an opportunity to learn.  We focus on the more important mathematics and cover a few less topics; however, the course provides more hope that all students can succeed regardless of their prior mathematics.

A central part of this fault-proof system is the instructor ‘assessing’ every student’s understanding in every class.  Work shown and dialogue reveal a much richer map of knowledge than can ever be achieved by technology such as homework systems.  Online platforms such as My Lab, Connect Math, and Web Assign play a role for students; however, they are not fault-proof — I believe that they tend to be even more ‘reinforcing existing achievement gaps’ than the basic traditional curriculum.

In general, the New Life model looks at the problem of equity by designing a curriculum that provides powerful opportunities to learn.  Our goal is to create a system where hard work will result in progress for every student.  Because equity is so important, we in the New Life project base our work on the value of instructors working with students on important mathematics in prolonged and intense ways.  No student should be blocked from success by the accidents of their prior learning experiences; no student should be blocked from considering STEM fields by faults in their mathematical knowledge.

If you’d like to see the talk by Uri Treisman, it is available on the Dana Center web site at http://www.utdanacenter.org/its-50-of-the-best-minutes-you-can-spend-to-get-a-detailed-examination-of-educational-inequality-in-america-uri-treismans-equity-address-at-the-nctm-annual-conference/

I hope that you will work with us to build a mathematics curriculum that avoids catastrophic failure, where every student can succeed in learning important mathematics.

 http://www.utdanacenter.org/its-50-of-the-best-minutes-you-can-spend-to-get-a-detailed-examination-of-educational-inequality-in-america-uri-treismans-equity-address-at-the-nctm-annual-conference/

Product As Sum: The Language of Algebra

I’ve been puzzling over some types of errors that seem both common and resistant to correction.  Essentially, the errors involve a disconnect between meaning and symbols especially in the two basic structures of quantities — adding and multiplying.

Here is a brief catalog of the errors:

• 3x²+5x² = 8x^4
• 4a(2b) = 8b + 4ab  (or some other ‘distributing’)
• (5y²)^3=15y^6  or 125y^8
• (3n +2) + (5n + 4) = 15n² +22n + 10
• sqrt(4x^9) = 2x^3
• sqrt(-50) = 5i + sqrt(2)

I’ve been seeing these types of errors for many years; however, it seems like the first 4 are becoming more common.  The radical context is not that important by itself for most of my students — except as a window into the same fragile knowledge about mathematical notation and meaning.  The errors appear with both new-to-college students and students who have ‘passed’ an algebra course.

In talking to students about these patterns, I’ve concluded that quite a bit of the problem is based on procedures removed from meaning.  Students usually know the phrase “like terms”, but seldom talk about counting when we have them; they know to combine the numbers in front but are often unsure about the exponents.  A focus on the meaning of the expression would make it clear what should be done.

The fourth error (‘foiling a sum’ or ‘distributing when adding’) is triggered by the “distributing is great” attitude; students really like to distribute, and we talk about distributing all the time.  In exploring this error (which shows temporary improvement) students say that they did not “see” the operation between the parentheses; what they mean is that they thought that parentheses means a product.

It’s likely that experienced teachers are not surprised by any item on the list above.  The issue for us is this: If these are important enough, how do we change our curriculum to decrease the frequency of such errors of meaning?  My own view is that the basic errors (the first 4) are very important, and I want to address them in all courses (whether traditional algebra or a math literacy course).

One strategy that I plan to use is more “unblocked practice and assessment”.  Much of a traditional developmental math course is severely blocked: the problems deal with a small set of procedures, separated from other types that might trigger an error.  We need to provide opportunities for these errors to be shown during the learning process.  Instead of trying to include quite so many types of each procedure, I will include some competing types from earlier work.  A student who can complete 50 ‘foil’ problems with 90% accuracy may not understand much at all, and may mis-apply the procedure … if we’ve never given them a chance to develop skills in discriminating types of problems.  This unblocked approach needs to be in all stages of learning (initial, practice, assessment, cumulative, etc).

Another method I use in my beginning algebra course is based on language learning concepts.  The idea is not complicated: Present students with either the symbolic statement or a verbal equivalent and ask them to identify the other.  Usually, this is done in a ‘multiple-select’ format: more than one correct choice is possible.  Students need to know that there is more than one verbal statement for a symbolic statement, and that there are sometimes equivalent symbolic statements.

For years, I have included some vocabulary or concept questions on daily quizzes.  I am concluding that I need to expand this to other assessments including tests, and to include perhaps more types.  Some of the online homework systems we use have these types of items, and the students who need them the most tend to skip  them … putting more emphasis on these in assessments will encourage students to take them more seriously in the homework.

I called this post “product as sum” because I am seeing students not being able to consistently treat them accurately.  This is such a fundamental concept that such errors bother me, especially when they occur in students who have passed an algebra course last semester.  Perhaps this is more evidence that:

1. We are trying to ‘cover’ too much (not enough time to understand and connect knowledge)
2. We focus on procedure too much (removes meaning as a critical feature to deal with)
3. We compartmentalize content too much (problems tend to be blocked, sometimes severely)

Meaning, connections, and concepts are important.  Procedures by themselves?  Not so much!

 Join Dev Math Revival on Facebook: