Archive for the ‘MathED’ Category

Signed Numbers: Colored Counters in a “Sea of Zeros”


The “colored counter” method is an old tried-and-true method for teaching the concept of adding signed numbers.  However, to show subtraction with the colored counter method has always seemed painful to me … that is, until I altered the method slightly.

Now all problems are demonstrated within a “Sea of Zeros” and when you need to take away counters, you can simply borrow from the infinite sea.  Voila!  Here’s a short video to demonstrate addition and subtraction of integers using the “Sea of Zeros” method.  You can print some Colored Counter Paper here.

Video: Colored Counters in a Sea of Zeros

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Giving up Calculation by Hand


This is scary stuff for math professors, but with the arrival of amazing programs like Wolfram Alpha, we’re going to have to start paying attention to the signs of change.  I talked to Conrad Wolfram (at Wolfram Alpha Homework Day) when he was still formulating what he wanted to say at this TED Talk.  I think it’s worth 18 minutes of your time to watch Teaching kids real math with computers.

Here’s an outline of the Conrad Wolfram’s argument (which I am paraphrasing/quoting here):

What’s the point of teaching people math?

  1. Technical jobs (critical to the development of our economies)
  2. Everyday living (e.g. figuring out mortgage, being skeptical of government statistics)
  3. Logical mind training / logical thinking (math is a great way to learn logic)

What IS math?

  1. Posing the right questions.
  2. Convert from real world to mathematical formulation
  3. Computation
  4. Convert from mathematical formulation BACK to real world

The problem? In math education, we’re spending about 80% of the time teaching students to do step 3 by hand.

Math is not equal to calculating, math is a much broader subject than calculating.  In fact, math has been liberated from calculating.

Should we have to “Get the basics first”?  Are the “basics” of driving a car learning how to service or design the car?  Are the “basics” of writing learning how to sharpen a quill?

People confuse the order of the invention of the tools with the order in which they should use them in teaching. Just because paper was invented before computers, it doesn’t necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics.

What about this idea that “Computer dumb math down” … that somehow, if you use a computer, it’s all mindless button-pushing.  But if you do it by hand it’s all intellectual.  This one kind of annoys me, I must say.  Do we really believe that the math that most people are actually doing in school practically today is more than applying procedures to problems they don’t really understand for reasons they don’t get? … What’s worse … what they’re learning there isn’t even practically useful anymore.  It might have been 50 years ago, but it isn’t anymore.  When they’re out of education, they do it on a computer.

Understanding procedures and processes IS important. But there’s a fantastic way to do that in the modern world … it’s called programming.

We have a unique opportunity to make math both more practical and more conceptual simultaneously.

Personally, I’m all for it.  But how?  That’s the question.  How to shift and incredibly complex and interconnected system of education? How to train tens of thousands of teachers and faculty to teach a new curriculum that they themselves never learned?  Hmmm … it seems that we might need some help, maybe a new paradigm for education itself.  It’s coming.

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Measuring Teaching and Learning in Mathematics


This weekend at AMATYC I presented this Prezi presentation on How can we measure teaching and learning in math? My husband was kind enough to act as the videographer for the presentation, and so I can also share the video presentation with you today.

I think the video should add quite a bit of context to the presentation, so I hope you’ll take the time to watch it.  What I propose (at the end) is a research solution that would help all of the math instructors in the country (who want to) participate in one massive data collection and data mining project to determine what actually works to improve learning outcomes.

If you have any suggestions for where to go from here, I’d be happy to hear them.

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Student Conceptions of Mathematics


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Do you ever get the feeling that you’re not making any real progress with your students?  Sure, they pass tests and progress through the courses (well, most of them), but have you ever just had the uneasy feeling that they really don’t get what math is all about?  Suppose you were to ask your students the following question:

Think about the math that you’ve done so far.  What do you think mathematics is?

What do you think they would tell you?

Well, in 1994, a research group from Australia did ask 300 first-year university students this question (Crawford, Gordon, Nicholas, and Prosser).  They identified patterns in the responses, and classified all 300 responses into categories of conceptions in order to explore the relationships between (a) conceptions of mathematics, (b) approaches to learning mathematics, and (c) achievement.

If you want all the gory details about the study, you’ll have to track down the journal article, but let me attempt to summarize their findings.

Students’ conceptions of mathematics fell into one of five categories (Crawford et al., 1994, p. 335):

  1. Math is numbers, rules, and formulas.
  2. Math is numbers, rules, and formulas which can be applied to solve problems.
  3. Math is a complex logical system; a way of thinking.
  4. Math is a complex logical system which can be used to solve complex problems.
  5. Math is a complex logical system which can be used to solve complex problems and provides new insights used for understanding the world.

The first two categories represent a student view of mathematics that is termed “fragmented” while the last three categories present a more “cohesive” view of mathematics.  Note that the terms fragmented and cohesive are well-used throughout the international body research.  The categories above, as you may have noticed, form a hierarchical list, with each one building on the one above it.

Here’s where this study gets (even more) interesting.  The researchers also asked students about how they studied (changing the language slightly here to “americanize it” a bit):

Think about some math you understood really well. How did you go about studying that?  (It may help you to compare how you studied this with something you feel you didn’t fully understand.) How do you usually go about learning math?

Again, the researchers went through a meticulous process of categorization and came up with five categories (Crawford et al., 1994, p. 337):

  1. Learning by rote memorization, with an intention to reproduce knowledge and procedures.
  2. Learning by doing lots of examples, with an intention to reproduce knowledge and procedures.
  3. Learning by doing lots of examples with an intention of gaining a relational understanding of the theory and concepts.
  4. Learning by doing difficult problems, with an intention of gaining a relational understanding of the entire theory, and seeing its relationship with existing knowledge.
  5. Learning with the intention of gaining a relational understanding of the theory and looking for situations where the theory will apply.

Again, these five categories were grouped, this time according to intention, into two general categories: reproduction and understanding.  In the first two approaches to learning math, students simply try to reproduce the math using rote memorization and by doing lots of examples.  In the last three categories, students do try to understand the math, by doing examples, by doing difficult problems, and by applying theory.  Other researchers in this community have seen similar results on both general surveys of student learning and on subject-specific surveys and have termed this to be surface approach and deep approach to learning (see Marton, 1988).

Still reading?  Good.  Remember my first question? Do you ever get the feeling that you’re not making any real progress with your students? Let’s answer that now.

Here’s how the conceptions of math and the approaches to learning math correlated in this study (Crawford et al., p. 341):

conceptions_of_math

Did you catch that?  Look at how strongly conception and approach correlates.  It’s probably what you’ve always suspected, deep down inside, … but there’s the cold, hard, proof.

Of course, all this is not so meaningful unless there’s a correlation with achievement.  At the end of their first year, the students’ final exam scores were compared to the conceptions and approaches to mathematics (again, for technical details, get the article). The researchers made two statistically significant findings:

  1. Students with a cohesive conception of math tended to achieve at a higher level (p < .05).
  2. Students with a deep approach to learning math tended to achieve at a higher level (p < .01).

Okay, so where does this leave us?  Well, we don’t have causation, only correlation (at least, that’s all we have from the 1994 study).  However, Crawford, Gordon, Nicholas, and Prosser were nice enough to use their research to develop a survey inventory that we can use to measure students’ conceptions of mathematics (1998).  The 19-item inventory (5-point Likert scales) has been thoroughly tested for validity and reliability, and can be found in their 1998 paper, University mathematics students’ conceptions of Mathematics (p. 91).

Suppose you want to try something innovative in your math class, but you don’t know how to tell if it works.  You could, at the very least, try to measure a positive change on your students’ conceptions of math (there are other ways to measure the approach to learning, but this is already a long blog post and you’ll have to either wait for another week, or view my presentation How can we measure teaching and learning in math?).

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To give the Conceptions of Mathematics Questionnaire (CMQ) would take approximately 10 minutes of class time (you should ask for permission from Michael Prosser before you launch into any potentially publishable research).  This would give a baseline of whether students’ conceptions are fragmented or cohesive.   If you were to give the survey again, at the end of the semester, you would be able to see if there is any significant gain in cohesive conceptions (or loss of fragmented conceptions).

So, I have permission (I met Michael Prosser this summer when he was at a conference in the U.S.), and I’m going to use this in all my classes starting next week.  I figure that when I start to see major differences on those pre and post-semester CMQ inventories, that I’m doing something right.  If I’m not seeing any gain in cohesive understanding of mathematics, then I’m going to keep changing my instructional practices until I do.

Papers that you will want to find (and read!):

Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1994). Conceptions of Mathematics and how it is learned: The perspectives of students entering University. Learning and Instruction, 4, 331-345.

Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1998, March). University mathematics students’ conceptions of Mathematics. Studies in Higher Education, 23, 87-94.

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How can we measure Teaching and Learning in Math?


Last week I prepared a new presentation for the MichMATYC conference based partially on the literature review for my dissertation.  In my dissertation I am studying instructors, but in this talk I addressed both the instructor and the student side.  It was also the first presentation I’ve built using Prezi, and it was interesting to re-think presentation design using a new tool.  Of course, the presentation misses something without the accompanying verbal descriptions, but there is enough information on here that you can begin to understand the problem (we don’t actually know much) and the solution (common language, common measurement tools).

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There are also a few new cartoons/illustrations in this presentation.  I’ve started just paying for a couple of illustrations per presentation to help viewers to understand (and mostly to remember) difficult concepts.  Just to give you a rough idea in the time involved to create something like this, I spent about 18 hours on the Prezi build (which doesn’t even begin to account for the time spent doing the research).

How can we Measure Teaching and Learning in Math?

If you attended the talk (in the flesh) and would like to evaluate the presentation, you can do it here on SpeakerRate.

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