Today, at the request of my CT, I pulled aside a small group (4 students) to try and see if I could help them begin using more advanced ways of making change. The four students I pulled aside were from a range of ability. One counted her quarters “25, 36, 82, 100.” Another was at the level where he would be comfortable making change. (”I want to buy your pen, which costs $0.46; if I give you $0.76, what change will I get back from you?”) The other two students were in the middle. I was hoping that by taking a range of levels, I could create an environment where the students could learn from and help each other.

I began by counting quarters and identifying values with the group to make sure that the basics of money were understood. Three of the students were great, the fourth still needed some work. I proceeded to give each student a “wallet”–a bucket full of play coins. I had the students look through their wallets and gave them a minute or so to interact with the manipulatives. I then showed them a book, and said, “Hmmm…I want you all to buy this book, which is 36 cents [at this point I wrote 32 cents on the board]. Can you make 36 cents from your coins?  Think about it, and give me a thumbs up when you’re done.”

After the students were finished, I went around and had each child show the group how they counted their coins.  I recorded their responses using number lines.  I represented the coins with jumps on the number lines.

•  Student One made 36 cents using:  three dimes, six pennies.
•  Student Two made 36 cents using: one quarter, eleven pennies.
•  Student Three made 36 cents using:  one quarter, one nickel, six pennies.
•  Student Four made 36 cents using: one quarter, one dime, one penny.

After representing all of the number lines on the board, I asked the students to help me count the number of coins in each method.

• Student One:  9 coins
• Student Two:  12 coins
• Student Three:  8 coins
• Student Four:  3 coins

Since I wanted students to begin using more “efficient” ways of putting together money, I pointed out how Student Four’s method only used three coins.  “Wow!  How neat if we can find the way that uses the fewest amount of coins!  This would be very fast, if you were in a hurry, to count three coins instead of twelve!  Let’s practice this!”

I pulled out another book and told the students that this book cost 58 cents.  “Show me how you can use your coins to buy this,” I said, writing the number on the board.  I allowed the students to work for a few minutes and when it looked like they were all satisfied, I asked them if they had found the way that used the least amount of coins.  Many hadn’t, and had relied on their previous, penny-centric method.  I told them I would give them a hint, “I want you to try to find the way that uses just 6 coins.”  This presented a problem for the children.  Some protested that it was too hard, and so I told them to work together and see if they could figure it out using teamwork.

After some trial and error, one student realized that you had to start with the biggest number (the quarter).  The students used one quarter, but hadn’t yet thought to use two.  I let them keep working, but redirected when I saw them struggling with the math.  One girl in particular could not add and keep track of such numbers.  She would pull out a quarter and a two dimes and count 31 cents (25, 30, 31).  I continuously held up various coins to reestablish their value, and showed children how to add on ten repeatedly as well.

I was very proud when the children finally came to the conclusion that the solution was to select two quarters, one nickel, and three pennies.  I gave them additional examples and found that the main inhibitor to success was the students’ ability to add the amounts.  The main reason so many reverted to pennies was because that is how most of them are currently adding–by counting on units of one.  Skip counting, as money is counted, is not a mastered skill for most of these students.

My initial reaction is that these students are not ready to be dealing with money.  Without the ability to comfortably skip count by tens and fives, this is nearly impossible.  However, while writing this, I have come to think that maybe money could be another way for children to access this information.  It seems that often, our natural reaction is to simplify information when our children appear to be struggling with it.  But maybe this is the wrong approach.  My students could drill addition strategies from now until June  and perhaps learn them, perhaps not.  However, everything in my training says that we need to push children with authentic meaningful assignments if we hope to help them succeed.  Perhaps money should be introduced first, rather than second.  Maybe money is an authentic pathway towards skip counting, and more advanced mathematics.  This activity was very eye opening for me, as it forced me to reevaluate my initial instinct.

Farmer Bob raises cows and chickens on his farm. One morning, he walks into the barnyard and counts 18 legs. How many cows and how many chickens are in the yard that day?

The first is inteded to prepare students for the pending problem-solving session; the second is to be used during the session itself; the third helps students develop metacognitive awareness and demonstrates that thinking continues on after the solution of the problem posed.

BEFORE: Begin With a Simple Version of the Task

As I read this chapter, this suggestion struck me as being particularly effective for my teaching scenario. I was a little concerned with the students ability to solve such a difficult problem. However, this presents the perfect introduction. By removing one variable, and only asking about the number of chickens in the barnyard, I can make the problem much more accessible to students. This preliminary activity will mentally prepare my students for the types of thinking they will be asked to engage in during the more intensive part of the problem-solving session. It also draws attention to the fact that there are two variables to consider when presented with the “real” problem. In addition to helping my students evaluate the problem, it allows me to evaluate the students. How are they able to grapple with this problem? What challanges does the inital, “simplified,” version present?

During this introduction, I think the teacher should be both modeling and scaffolding the simplified version of the problem so that students are prepared when “turned loose” to grapple with the more complex situation. Ideally, in my mind, ideas and suggestions should be recorded on the board, without judgement or praise. By refraining from such reactions, the instructor is able to elicit the widest possible array of answers. These varied solutions (even those that are incorrect) could be an incredibly useful reference tool as children begin to experiment and test their ideas. To be sure that all answers and theories are recorded correctly, it is imperative that the teacher ask lots of follow-up questions, rather than summarize what (s)he thinks the student is saying.

DURING: Encourage Testing of Ideas/Find a Second Method

Although these are listed as two different bullet points within the chapter, they complemented each other so well that I felt compelled to point out both, here. When conducting a problem-solving session, it is very important to be able to let go and allow children to investigate any avenue that might occur to them. Teaching them to trust and investigate their own ideas is a powerful skill that will carry them through all endevors, both academic and otherwise. However, the teacher cannot simply disappear during this investigation. Rather (s)he must be available both to encourage and challenge.

For the problem posed above, these two methods work together to do just that. Children, who are very perceptive and have picked up on the fact that there are “right” and “wrong” answers in math, often want to know if they are doing the problem right. Rather than providing encouragement or suggestions for improvement, it is important to really let go and allow their instincts run the direction of the investigation. Encouraging the testing of ideas is one way to remain neutral. If a student asks, “Is this the right answer?”, the teacher should counter with, “Well, let’s see. Can you think of a way to test that and prove if it’s right or not?” If the students are particularly independent, it can still be effective to remind students to see if their thoughts make sense. One possible way to do this without making a value judgement is simply to remind students to explain their thoughts, not merely the steps.

If the students are having trouble testing their answer (or if they solve it incredibly quickly), a great strategy is to ask that they find a second method of doing the problem. This suggestion is particularly effective as it works in two different ways, depending on the student’s level of understanding. If a group (or student) is having difficulty finding a way to test an answer to see if it is correct, finding a second way of doing the problem can help provide a proof. In this way, the suggestion works as a scaffolding tool. However, if the group/student solves the problem quickly and is at a more advanced level of understanding, this same suggestion turns into a differentiation tool to provide additional challenges.

AFTER: Identify Rules, Hypotheses, and Future Problems

I found the suggestion to “identify rules, hypotheses, and future problems” particularly interesting as a way to conclude the lesson. It’s important (for both the teacher and the students!) to remember that thinking does not cease with the solution to a problem. Any scientific investigation should end with the generation of a new set of questions, and asking the question is often the hardest part of any problem. So the goal should not be, “What was the answer to that problem?” but rather, “What implications does this have?” Additionally, the process of sharing information and participating in a whole-group approach to inductive reasoning teaches students how to become active learners in their communities. They begin to develop a critical approach to discourse, where evaluation and theorizing can occur.

The teacher must once again work to maintain a neutral stance during this process. Refraining from commentary may initially stifle discussion, but if practiced continually, students will begin to provide answers without constant affirmation. This puts the focus of the discussion on the process rather than the result of inquiry. Suggestions (incorrect or not) without any rationale behind them should be recorded as “[Student's] Hypothesis.” At this point, the floor can be opened, and further investigation can occur as classmates struggle to prove or disprove the theory. Even if actual hypotheses never occur, such discussions can still generate some fascinating questions. The teacher should carefully listen to explanations to identify any questions that my be formulated, though never explicitly stated.

I feel that as a whole, this chapter has been very helpful in providing a framework within which I can develop my own problem-solving lesson. I look forward to testing the practical application of these suggestions in the near future!

My math CT opens class with a game that she calls “Squash.” She chooses (or more likely decides as the game goes on) a secret number between two points, draws a relevant number line, and asks the children to guess her secret number. When they guess, she plots their point on the number line, tells them if her number is greater or less than the one she chose, and shades out the eliminated, “squashed,” portion. When the children finally reach the correct number, she asks them what they can tell her about the number. What is special about it? What is it made of? Etc.

Many times during this game, I’ve heard her “reinterpret” a child’s statement so that it was correct, when the actual wording was more ambiguous. One of the most common mistakes that I see is when children explain how to break down a number, for instance, 43. Many children will say that 43 is really “four tens and one three.” My CT quickly corrects this mistake (seemingly subconsciously) by repeating, “Okay, so here we have four tens, and three ones.” Though it seems like a small mistake, these are very different ideas!

I want to see if children really understand the base-ten place value system, or if they have merely learned the “procedure” of how to respond to her request. I plan on asking the following questions:

• How does the game “Squash” work?
• How do you choose what number to pick during Squash?
• Why do you pick this number instead of that one

Afterwards, we will play one round of Squash. After each guess, I will ask:

• Why did you pick this number?
• How did you know it wasn’t going to be this other number?
• So if my number is (greater than/less than) yours, where should I shade in my number line? What have we ruled out?

After determining the correct number, I will ask the children to tell me what they can about it.

• Is there anything special about this number that you are aware of?
• Can we break this number down at all?
• How would you fill out a tens/ones chart using this number?
• (Flipping the chart so it is a ones/tens chart, and giving a new number, I will ask:) What about this one? Can you do this one too?

Finally, I will draw a pictorial representation of a number containing tens and ones (rods and units). I will finish by asking:

• What number is this?
• How did you know?
• Show me how to read this.
• (I will draw a similar picture, but not draw the ten individual units in the rod.) What about this number, how would you count this?

Hopefully, this will help me ascertain the level of understanding that my students possess. This will help me immensely as I begin planning my lessons in the near future! In a sense, this is a sample set to help me pre-assess my class as a whole.

Surprisingly, this ties in with Van de Walle’s discussion on the two types of mathematical knowledge.  Most of my (and I’m assuming many others as well) experience with math has been purely in the form of what he calls procedural knowledge.  Procedural knowledge can be understood in terms of rote learning: carrying out a specific set of rules or procedures to achieve a given task.  This is why many instructors rely on nothing more than worksheets and repetition.  They are trying to drill the procedure in through rote repetition, which is effective in this type of situation.  Unfortunately, this provides no opportunities for the child to really understand the concepts of mathematics so that they can begin to build and modify schema.

Conceptual knowledge, on the other hand, is an understanding of the relationships between numbers, symbols, or ideas.  This is a much more difficult type of understanding to come by, as it undergoes an endless journey of adaptation, modification, and application.  Conceptual knowledge is never completely the same.  A rectangle can represent a whole, a half, a third, a quarter, or any imaginable value depending on its context.  Likewise, numbers exist within a specific context: a fairly universal context, but still a context.  Numbers exist within a base ten framework.  This means that all rebundling, regrouping, and breaking apart occurs with the assumption that the “whole” is a unit of ten.  One square into ten rods.  One rod into ten blocks.

With this in mind it no longer becomes surprising that I only understood the absolutes of mathematics.  Many teachers spend so much time on procedure that there is no time for conceptualization.  However, in order for our students to become truly fluent in the world’s “universal language,”  it is imperative that they learn to manipulate the relationships between the absolutes and recognize and make use of the negative space that exists.   The issue of negative space could lead into some fascinating discussions and lessons for upper-el students!

At this point, I am completely on-board and excited about the richness that conceptualized understanding could bring to mathematics.  However, I am unsure as to how to weigh this against procedural knowledge, which must still be covered.  I am nervous about teaching math for precisely this reason.  I want my children to understand and really love math in a way that I never could; unfortunately, there are certain standardized checkpoints that I must meet in order for them to be successful in future classrooms.  With such limited time and resources, how can I be sure that my children have opportunities to build conceptualizations, frameworks, and schema?  Also, what is the best means of instruction?  Didactic, inquiry-based, or idea-based?  Why does this model have value over the other two?

I have had great success with my courses in the TE department thus far (though I cannot say the same for required courses in other departments). I find that my TE courses have been wonderful at giving me an opportunity to merge theory and application. I also appreciate the fact that the college recognizes that our ultimate goal is employment, and works to create opportunities for me to build my professional portfolio and tailor assignments to my own personal goals. With this in mind, I expect this course to follow in that tradition.

Personally, I am an advocate for critical pedagogy. I also believe that technology is an incredible resource that can be used to bring new opportunities to students who may lack social and/or cultural capital. Math is slightly vexing to me, as I’m not sure how this subject fits into my pedagogical views. I have a few examples of how a critical approach can be applied to math education that came from an article detailing the key components of a critical pedagogy. The teacher in one example was doing a unit on measurement, fractions, and fraction computations. Rather than give children a worksheet with various drills, she presented them with a legitimate project. The students (4th and 5th grade) were given the following problem:

We are making a bookcase to hold our new stereo. We need to have 3 shelves. The top shelf must contain 3 compartments; the second shelf, 2 compartments; and the bottom shelf, 1 compartment. We also have 6 boards that are 60″ long, 2.5′ wide, and 1″ thick. Draw a diagram of what the shelf will look like when finished. Using fractions, show how you will cut the boards to make compartments.

This is a very high-level investigation. There is no question of why this information is relevant. It’s immediately obvious. It forces the students to organize the information coming into them so that it becomes relevant. Finally, it demonstrates that math is not formulas and numbers, it is used in all aspects of our daily lives.

This is the type of math that I believe should be taught in our schools. I hope to create investigations that meet the three criteria of an authentic, critical pedagogy. First, that the assignments be carefully scaffolded so that knowledge is co-constructed by the students, rather than reproducing knowledge discovered by others. Second, that inquiry remained grounded in a field of knowledge, broadening the student’s schema.  Finally, that the student demonstrate achievement that has value beyond mere summative assessment.

I feel that our past few sessions have been a great introduction to the course, and have helped facilitate the creation of an effective learning community.  However, I have to admit that I am eager to get into the “meat” of the course.  It seems like what we’ve done up until this point is merely a review of the things that we should have learned in Math 201 and 202.  I am eager to move beyond these types of discussions.