Decimal Pickle

In our last class we played a game called Decimal Pickle.  All you need is a deck of cards with 10, Q, and K removed, and a pencil and paper!  It's very simple to set up-- draw a path of ten steps, you could use circles, arrows (like I did), or any shape you want.  The goal of the game is to create sort of a number line where you can place decimals in order on the path from 0 to 1.  The cards you flip over on your turn are the numbers you get to arrange to form a decimal to place on the path.  Red cards mean choose another card, up to three cards.  Black card means stop.  So for example I flip over a red Jack which represents 0, and a black 8 so I stop.  Now I have two possibilities for decimals: .08 or .80.  Since there are no blank arrows between .032 and .11 I cannot place .08 on my path.  However, .80 is greater than .789 and less than .938, therefore I can place .80 on my last blank arrow, and I win!  

A couple things to keep in mind, if you draw a decimal that is a repeat, say I draw a black 5 again, I cannot fill in two spaces with the same number.  Also if there's no space for the decimal I get I must pass on that turn, for instance if I draw a red 1, and a black 2, the only possibilities are .12 or .21 which would not fit between any existing arrows, so I pass.  

I think this is a great game to develop an understanding of decimal quantity.  I think it especially emphasizes quantity in terms of the decimal places.  For instance, it helps students learn that there's a big difference between .37 and .037 because each new decimal will have a context--it will have numbers less than and greater than the number.  I also think this could be a nice intuitive introduction to start learning inequalities and their symbols, or possibly review them if they had learned about them a bit in elementary.  

There is also a benefit in being a pair game because students can make mistakes in placements and the whole class won't notice, only their partner might.  So there's an aspect of not only knowing your game, but also checking to make sure your partner is playing correctly.  It provides lots of examples and nonexamples of appropriate placement of decimals and can involve all levels of understanding.  

For us college students it really helped try to break our habit of saying "point zero six seven" which really has no mathematical meaning and instead practice saying "sixty seven thousandths" which can help set the context of decimals as fractions of tenths, hundredths, or thousandths.  Pronouncing decimals as fractions can help students with the fluidity of switching between the different representations.  If a student was given the fraction 7/10 they would have a much easier time creating the decimal .7 if they were used to hearing .7 as seven tenths instead of point seven.  

There are some variations that can make the game more fun.  My partner and I decided to increase our paths to 15 spots so it would not only take longer to fill in, but it would also make it more difficult to know how to space the decimals.  Instead of having the strategy of putting .5 in the middle, you now no longer have an exact middle.  So in your first couple turns when you can place your decimals in any of the blank spaces, you have to think harder about what numbers could eventually go in the spaces you leave blank.

I love strategic games, and I think this is the perfect combination of a simple game concept that can really aid learning while still having enough strategy and variations to make it fun and exciting for all types of learners.

Levels of Understanding: Relational Vs. Instrumental

My post today is a reflection regarding the article called Relational Understanding and Instrumental Understanding by Richard Skemp.  First, a quick summary.  In this article Skemp discusses the two types of understanding, one could even call them levels of understanding.  Relational understanding he defines as "knowing what to do and why," with an emphasis on the why aspect.  Alternatively, Instrumental understanding is knowing a rule and when to use it, but not why.  He calls it "rules without reason."

In the article he goes on to share a few different analogies.  One was quite helpful in which he drew a comparison to music.  He said instrumental understanding was like knowing the notes and the staffs on paper, but not having those figures attached to sounds.  Relational understanding was like learning the notes in conjunction with the sounds.  Consequently when asked to write a melody the students taught relationally, would have a far easier time writing a melody.  I fully agreed with this analogy; of course reinforcing the meaning of notes with auditory information would increase the fluency and skill of students writing music.  Music class would in intensely boring if nobody heard any music.

The analogy that I didn't agree with was when he compared understanding to two different ways of navigating a new city.  Instrumental understanding was compared to just getting directions to and from essential places, which can lead you to be incredibly lost when you make a wrong turn.  Relational understanding was like exploring the city on your own and creating a mental map so you can recognize multiple ways of getting from A to B.  While I appreciate his analogy on some level, I still think it's problematic.  His description of exploring the city to find your own routes relies on doing so without help from an expert.  It's purely self discovery.  However, I believe he wants teachers to be teaching the relationships and alternate routes to get the answer.  I'm not sure that I agree with the explicit teaching of these relationships though.  I think it's sort of intriguing when I don't learn every detail about a rule and then I find it out on my own when I continue learning about the subject.  It s almost like an incentive to learn more so you can get the satisfaction of learning why a rule works by yourself.  However, I am extremely intrinsically motivated, so I understand why not all students would have the motivation to look for connections by themselves.  But I also think it's a good life lesson to not spoon feed every connection to students.  They need to be able to recognize connections on their own.

As teachers perhaps we could set up questions that will inevitably lead to discovering connections, but we should leave it for them to fully explore.  For example when discussing slope, many kids mix up the slope formula and divide the difference of x coordinates by the difference of y coordinates (like x/y), instead of the correct y/x.  To aid the relational understanding of this topic, I could ask my students to compare slope formula to the well known trick "rise over run."  Hopefully they would make the connection that the difference in y coordinates is the numerator because it corresponds to the rise.  Similarly the x corresponds to the denominator because it is the run between the two x values.  So this quick examination of the slope formula and what it corresponds to on a graph is a simple way to build a connection that will help students remember the slope formula.  But I think it's important to not answer the question for the students, but rather pose the question and leave them to find the relationship.  I think the self discovery will create independence and hopefully build some intrinsic motivation to find the relationships in upcoming lessons.

Another thing that bothers me about Skemp's outright preference for relational understanding is it's inefficiency.  Even if there are a million ways to do something, I think there is something to be said about knowing how to do it with ease.  I think math can be explained in so many ways, but since it values efficiency, we should teach with a preference to the efficient way to answer questions.  If nobody valued efficiency, no formulas would be created.  Formulas are mountains of work that are consolidated so that we can use them quickly and accurately.  I think that idea is discounted in the relational understanding model, but it shouldn't be.  Our students don't need to reinvent the wheel, just generally learn the concept of it so it can help them later on.  

Overall I'm not completely convinced that explicitly teaching relational understanding is the best method.  I think instrumental instructions has its merits as well.  Relational understanding could be beneficial when students are completely struggling with a concept, or as an extra independent enrichment, but I don't think teaching instrumental tricks to help kids remember certain concepts is terrible.  I think there's a time and place for both levels of understanding.

Fractions

To introduce the dreaded concept of fractions, our class decided to do exercises that would push the subject and necessitate the learning of fractions.  One such problem that isn't easily explained without the use of fractions was the Mixture Blues problems, some of which can be found in the image above.  Basically, in each container there is either blue dye or water and you have to find out which combination is more blue, A or B.
This problem sparked quite the discussion about how to go about solving these problems.  Of course, as college students we knew how to turn each set into fractions with common denominators, but when trying to reason through it without using fractions, we all struggled a bit.
Some of the methods we attempted was comparing how blue they were in terms of how many they would need to be all blue.  So in problem B1, A would need 1 more blue and B would need 3 more blue, so A is more blue.  We also attempted to "cancel out" similar containers in each set.  So for B4, two blue of each set would cancel out as well as two clear, so the top would be left with 1 blue and the bottom would be left with nothing.  So we argued that set A with the one blue would be more blue than set B which has nothing.  This method, although seeming intuitive to some of us, caused lots of discussion amongst the class.  Especially in the following problem,

We canceled out three blues and two clears from each set, but then were left with nothing in A and a blue and a clear in B.  From our logic, B would probably be more blue since a blue and a clear could mix to be a half blue which is more than set A which has nothing.  However, when you compute the fractions, A is actually .03 more blue.  Our intuitive method of canceling-out failed.  But why?
Let's take problem B5 and see why canceling out doesn't fit with fractions.  A= 3/5, B=4/7.  Cancel out two blues so now A=1/3, B=2/5.  If we just stop here we can already see that we've violated the rules of fractions since 3/5 does not equal 1/3 and 4/7 does not equal 2/5!  Really, the only way we can solve this problem is by introducing fractions and being able to compare them with common denominators or by turning them into decimals.  So once we reveal problems that cannot be solved without fractions, hopefully students will be more motivated to understand fractions.  Without the context of fractions representing parts of wholes, it is difficult for students to comprehend the use of fractions.  And without the frustration of failing to consistently solve a real world problem, students will have little motivation to learn the sometimes baffling, but important, concept of fractions.