Math Hombre

@AlexKraker

Guest Post by Alex Kraker. This post is lightly adapted from Alex's teaching philosophy for his teacher assisting portfolio. Teacher assisting is a kind of half-time student teacher experience that our novice teachers do before a more traditional student teaching semester. I found this very uplifting, and he was willing to share with you all.

Teaching Philosophy
The teacher should be a beacon of knowledge, like a lighthouse, whose sole purpose is to shine their all-knowing cone of light round and round to each student and burn the desired and necessary knowledge into the eyes and brains of students.  The teacher holds all of the knowledge and it is their job to dole out the material and skills that are “necessary” to the students.  Nobody shall get to the knowledge, except through the teacher.

It seems as though too often we as society, parents, and even students fall into the trap of believing that nonsense in italics.  The job of a teacher is not to teach at all; to me it seems like such a misnomer.  When I consider what I need to, want to, and should be doing as a teacher, I feel like I am much more of a facilitator than a teacher.  I don’t want to be standing up in the front of the classroom mindlessly droning on about what a y-intercept is…that’s not my job!  My job is to be on the front lines, fielding questions, guiding inquiries, and motivating students to discover and learn all these new, wonderful ideas that they have yet to encounter.  I should be much less of a teacher and more of a tour guide.  I should be pointing out things that students may not have noticed, give ideas on what they could try to help solidify understanding, and challenging them to do, not just learn.

I feel as though students do not learn well when someone is just imparting knowledge to them.  Lectures are boring.  Students struggle to pay attention and get all of the material when it is just being thrown at them.  The best learning comes when students get their hands dirty.  When they’re given a question they can’t yet solve.  When they have to think about what they already know and how they can use it to find out what they need to know, that’s when learning occurs.  I believe learning is not a linear process.  Acquiring new knowledge always seems to come first, but that’s not real learning.  Learning occurs when we assimilate our knowledge, make connections, and understand what we just found out.  Learning is so much more than just finding something out that we didn’t previously know. It is a process, we acquire new knowledge, and build upon that.  Then once we are comfortable with what we have just figured out, we build more on that, so on and so forth until we go from just a few simple ideas to a whole web of knowledge, connections, ideas, and discoveries.  That whole process is learning, and that web represents our progress.

There are a few necessary components to learning.  I feel in order for learning to occur, students have to be engaged, involved in discovery, and entertained.  I’m not trying to say school always has to be fun, but it is so much more difficult to forget something that you enjoyed being a part of discovering.  And when you realize how you discovered it, you can go back through that process and discover it again if you forget specific facts.  Memorizing formulas for the volume of three dimensional figures isn’t learning.  Using manipulatives and two dimensional formulas to figure out how we derive the three dimensional ones leads to students being able to rediscover the exact formulas themselves if they forget what exact numbers we use.  It is my job as a “teacher” to help students understand why.  However, my job doesn’t stop there, it doesn’t even start there.  I have to help them care; see why it’s important, how we use our material and how it can help them.  I need to help them figure it out, answer questions, guide thinking and discussion, encourage participation.  I have to help them sort out what they just answered, see where it comes from, how it connects and what it can do for them.  My job is not to impart knowledge, but to feed the desire to learn and know.

As such, I must always be thinking of new ways to inspire, motivate and make students care.  I must identify areas where they may struggle, or things that can cause roadblocks in our journey to know more.  I have to be prepared for anything and everything and really know my students.  My job should never be the same from year to year.  I need to constantly adapt to my students.  I need to learn how they learn, know what they know, struggle to find where they will struggle.  Different students will need different types of instruction.  Leading them to discoveries will work well for some students, but other students may struggle with seeing connections between what we are doing and why it is important.  For some students, I will have to take a more direct approach.  I will have to simply teach them some things, and work on making connections once they feel comfortable with the material.   Not every student will be motivated, so I will have to find a way to motivate them outside of grades or the simple pursuit of knowledge.  Sometimes I will have to simply fall back on the old expectations of a teacher, and I will have to lecture on occasion.  However, it is my job to never fall back on lecturing and simply trying to force the knowledge from my head to theirs.

I want to shake things up.  I want my students to look forward to my class.  I want my students to feel like they are teaching themselves, like they are the catalyst for their learning.  I am looking forward to the challenge, and I am up to the task.  My teaching philosophy is that I am not a teacher, but so much more.

Image credits: ~John~ & JTKnull @ Flickr

At our university, content educators are mostly in their respective content departments, which is why we have a dozen or so math educators in our math department.  In our secondary teacher prep, we have three courses that are our "Math Ed" courses: Math 229, which is HS content focused, Math 329 - which is MS content focused, and Ed 331 - which is our content seminar for teacher assisting, when the novice teachers are in schools for the mornings and teaching at least a unit.  We are in negotiations to see them during student teaching, which will be excellent.

This is another guest post from a student assistant: Brock Walsh.  He paused school for a bit, but then came back with a very clear motivation about wanting to be a teacher.  Dave Coffey already posted a bit from him, where he used the NCTM process standards as an outside resources.

In his teacher assistant portfolio, he reused a bit from his 229 class, and I thought it was a neat opportunity to follow a student from early on until later in their teacher education. As an add on, I also included his piece from this past semester on the Conditions of Learning, which Dave recently posted in the Learning Museum.

Equity - Insights from the Past

(The following is a paper that was written for MTH 229, in which I had looked into the principle of equity as I related it to my experience of a nine week observation.)

Articles: Excellence in the high school classroom is something that teachers strive for. Sometimes conducting a learning filled classroom can be easy, but other times a teacher might not fully see and take advantage of teachable moments for all students. Being aware of and preparing for these teachable opportunities for all students to learn at a higher cognitive level is vital and defined under the Equity Principle of the high school principles and standards.

The article “Focusing on students’ Mathematical Thinking” by M. Lynn Breyfogle and Beth A. Herbel-Eisenmann focuses on trying to understand the thought processes of a student’s reasoning instead of relying on a student’s answer. Reasoning occurs when a student has time to think and then explain their thoughts. The time that is given after a question and before an answer is known as “wait time” and within this time, a student’s cognitive thoughts will increase. In the article, the authors emphasize an important detail. They quote from their findings, “Although most teachers are aware of the importance of waiting after they have asked a question, the importance of waiting after a student responds has received less emphasis (Rowe 1986). This all relates to students maximizing their learning by having the time given to them so they can process ideas for themselves.

The article goes further to say that when a student has given a correct answer, we as teachers should question them as to how they arrived at that. Students will often learn the most from themselves or when another student explains their reasoning. Asking for justification is a great way to evaluate not only a student, but the class as a whole when they respond and get involved in the discussion. Putting both “wait time” and “justification” together strongly represents the idea of equity and its importance in class.

The article “Unveiling Student Understanding: The Role of Questioning in Instruction” by Azita Manouchehri and Douglas A. Lapp relates to the Equity Principle directly by emphasizing the point that we as teachers need to ask the right questions for optimizing a lesson. Our questions need to facilitate learning and with the right questions being asked we can pull out conceptual reasoning from the entire class.

Magoo0311 @ Flickr

Personal: The only class in high school that ever truly challenged my reasoning was AP Calculus. Not because it was a hard class, but because the teacher invested so much into our learning and asked questions that forced us to explain ourselves. The mathematics classes that I have taken in college act the same way. The professors ask questions that require my justification. Sometimes I don’t fully know how to justify my answer and that can be blamed on the fact that I never had to do it through grade school. The in-class illustrations from the articles represent teachers asking questions that facilitate class, but do not emphasize reasoning like the classes that I have taken in college.

One specific instance of equity that I can remember my AP Calculus teacher applying was related to group work. The class was split into groups of fours and had to present on asymptotic behavior. Each person in the group had to focus on one specific aspect to present to the class and the groups had to hold each other accountable for their work. I can remember that there was a ton of questioning that occurred which felt like a debate. Within that debate, a lot of reasoning was taking place and uncertainties were being explained! The class as a whole was involved, and that is something special when an entire class is participating in discussion. In general, any time a teacher is at the front of a classroom instructing or going through a worksheet, and maybe only asking questions that a few students answer is a case of poor equity and should be avoided.

Observation: I conducted my observation at a well funded school with nice facilities. I observed a teacher and her freshmen/sophomore Geometry class during sixth period on Tuesdays and Thursdays. The class was primarily of white ethnicity, but there was one black boy and girl, and a hispanic girl. There were 15 females and 12 males in the class. The three learning objectives that I observed were review on algebraic properties, theorems about angles, and the last day was devoted to preparing for an upcoming test.

The questions posed in class were probably 50/50 for being open or closed. I noticed that when a question was presented in open form, there would generally be justification with the response. I compared notes with Susie K. from class to find a comparison between in class questioning. She told me that from her observation at Jenison High School, students were asked open questions about half the time in an algebra class but in the geometry class there were generally more open-ended questions. This makes sense to me as it seems fit that the higher level class should be challenged by the questions they get asked. A teacher should expect that as a student’s cognitive level grows; then they should also be able to reason more in depth. The wait time in the class I observed was generally around 3-4 seconds. This is reasonably good, but like the article proposed, there was really no wait time after a student responded. A good way that the teacher made sure each student in the class would have time before a response was by saying, “Everyone think about the problem by yourselves and then compare with a neighbor.” This way, students have time to learn by themselves and from their peers.

Further identifying the questions asked in class, about 22 percent of them required justification. I consider this to be a relatively adequate amount for a Geometry class, but would be something I would like to see get higher in preparation for more advanced math classes. A good example of an open-ended question that required justification was, “What justification do we get for AB+BC=AC?” I realize this seems obvious but sometimes that is exactly what is needed. She also asked, “If both L1+L2=180 and L2+L3=180, then shouldn’t they equal each other? Explain how you know this.” This question set-up made the students think about the properties and theorems that apply to these statements. These types of questions force students to think about possibilities. When called to answer, then the student can explain their best reasoning for an answer. Justifying yourself will sometimes correlate directly with equity if an explanation is clear and insightful so that the whole class learns from the response to the question. All of this stems from the question though, if a good open-ended question was not asked to begin with, then the opportunity for and learning in general has been lost. A good open-ended question posed in class was, “When I say adjacent angles, can you picture that in your mind?” Another one was, “How do you prove something true? What does it take to accomplish this?”

Interview: I conducted my interview questions by simply asking a few questions after each class to get a general sense of what she expects form her students. About instruction and questions Cristina said, “When I generally ask questions, I expect the students to think before giving a response. I’ll ask for understanding from the entire class and if nobody has a question then we move on. I expect students to ask if they don’t know. For communicating, specifically in Geometry, I expect students to use the correct language and correct theorems/properties. It’s important for students to have this foundation.” For the workload she said, “Homework happens every night, and each student is expected to complete their work or at least give a good attempt towards answering the question. Of course, I want all of my students to do well. It is really up to the student to provide the effort, and I am here to help each individual student as much as possible.”

Outside Resource - Conditions of Learning

One Laptop Per Child @ Flickr

This is my opportunity to share my understanding for the “outside resources” portion of my portfolio. During the exit interview, I was asked to explain my reasoning for using the Process Standards from NCTM, and Cambourne’s Conditions of Learning. Somewhat confused by this inquiry, I responded that I included them because they are both a framework that I feel needs to be implemented in the classroom everyday. These are both a resource that I want to keep a focus on when I teach because when using them, I feel my learners will effectively learn more. I was told that these were not the usual types of resources that are used, but upon my explanation, John and Dave understood my intentions of having them included and commended me for seeing these outside resources as a means for having a framework that benefits me in the classroom. Including them in this portfolio is a way for me to have a constant reminder of them.

Holy cow, have I been busy.  Sorry for the lack of new posts.  What makes it worse is that I have had several guest posts to get up that students were kind enough to send me weeks ago.  In addition to scads of things that I want to write up for myself!


The first is from a preservice secondary teacher named Jill Beauchamp.  She is active in coaching cheer, and in our local Dutch culture.  (And it's almost tulip time.)  I'm pretty sure she's a licensed wooden shoe dancer.

On an assignment that gave a choice of follow up options after playing with Pierre Van Hiele's mosaic puzzle in class (from “Begin with Play,” by Pierre van Hiele, Teaching Children Mathematics, Feb 1999), Jill chose to make a activity based on my triangle puzzle.  And she was willing to share it!  I like how she really captured Van Hiele's idea of beginning with play, and uses the puzzles to get at the triangle properties.  She makes the most of what I was designing the puzzle to do, have one triangle of each type.



The assignment:

Teaching Math – Mosaic Making

Choose one or more of the following to do for this:

1. Analyze Van Hiele’s mosaic. What geometric properties of the pieces permit all the combinations we saw in class?
2. Create your own mosaic puzzle and document your design process.
3. Create a new lesson using PvH’s mosaic or my 7 triangle mosaic at http://mathhombre.blogspot.com/2010/12/triangle-puzzle.html

Document your work, and be sure to include a reflection.

Schema: I decided to take a look at your 7 triangle mosaic – nice work! This would be difficult for me to create on the computer, so I am very impressed. When first thinking about a lesson in regards to the mosaic, I could only consider it being a fun puzzle. With our exposure in class to different workshops regarding the original mosaic, I began to think about the properties each triangle in your mosaic possessed. You have:

• -Two right triangles
• -One isosceles triangles
• -One right isosceles triangle
• -One equilateral triangle
• -Two scalene triangles (One acute and one obtuse)

Fabulous! You have one example of everything.

Focus: With my class, I would want to explore why these triangles fit together the way they do. Assuming the students have not yet learned about triangles, this could be used as an introduction. Let’s say I have this class for 60 min. Here’s how my day would go.


Introduction: (5 min) Talk about puzzles!

• What kinds of puzzles do students like to do?
• What makes a puzzle puzzling?
• What are some mathematical properties of puzzles?

Introduce Mosaic

Mosaic Play and Record: (15 min) Allow students to play with the pieces and try to create the mosaic. As they do this, I would like them to document their actions:

1. What they tried
2. What pieces worked together?
3. What didn’t work together?
4. Qualities they notice about the triangles

**If students solve the mosaic, they should focus on:

1. Is there another way to solve it?
2. Why do some types of triangles fit together and others don’t?

Discussion: (20 min) I would ask all students to pull apart their mosaics and separate the individual triangles. Then I would ask them if they saw any similarities between any of the triangles?

*As this is happening I will write up student ideas on the board. If need be, they may come up to the board and illustrate their thinking.

Assuming they already know terminology for a line, angle, point etc. I will have students pull out the rulers and protractors to assist them in drawing more comparisons. Once we have a pool of properties, we can begin to group the triangles accordingly. Once we are able to do this accordingly by the deduced properties, I will write the names of the triangles on the board (but not yet with their corresponding group). Instead I will ask students what they think goes with each.

Properties: (With any luck, we get some or all of the following, although I’m sure I’ll get some other interesting thoughts!)

• 3 equal sides
• 2 equal sides
• No equal sides
• 3 equal angles
• 2 equal angles
• No equal angles
• Right angle
• Obtuse angles
• Acute angles

Hopefully, they will see comparisons between the word “Equilateral” and the same angle and side measures, “Right” and the triangles with 90 degree, or right angles, “Scalene” and the triangles that depend on their individual scale/measure, although “Isosceles” doesn’t work too well, but it can be the odd guy out.

I will want to pay special attention to that sneaky little purple “Right Isosceles Triangle.” This guy is important because he shows that two properties can hold for one triangle. Maybe we could explore which properties can hold together and which ones don’t (As I’m writing this these ideas are just kind of coming…)

• A scalene can be a right triangle. Why? Because one angle may be 90 degrees, the other two differing, and all sides of different lengths. A scalene cannot be isosceles or equilateral because it goes against the definition of scalene.
• An isosceles triangle can also be right, but can an equilateral triangle also be isosceles? No, the definition of isosceles is EXACTLY two sides of equal length. Although it can be either acute or obtuse depending on the size of the angles
• A right triangle can then be isosceles or scalene. It cannot be equilateral because one angle must be 90 degrees, thus going against the fact that all angles in an equilateral triangle must be 60 degrees.

Teacher Question: So then, are triangles actually right triangles? Or does the word “right” just classify a specific type of isosceles or scalene triangle? A right triangle cannot exist outside of one of the two classifications.

Sorry for my tangent. The above discussion over the “right isosceles triangle” may be something for another day! My hope would be to get to the last part of my lesson…

Discovery: (15 min) The students would then need to reassemble the mosaic (I will show them the put together puzzle if they need it). With their protractors and rulers I would like them to work on:

Measuring the divided angles in the corners of the square. What is the sum of these angles? What type of triangles have an angle like this?

Measuring the divided angles along a straight line within a puzzle. What is the sum of these angles? What do they notice about all of these sums along a straight line? How does this compare to the sum of the angles within a triangle?

Lastly, I would like them to paste their mosaic together on a piece of paper and write out the angle measures, side lengths, and classification for each triangle. Students should make a note of anything else they notice.

Reflection: (last 5 min of class) What is one realization that surprised them today? Can they put anything they’ve seen into another context? How might it relate to something else?


MY Reflection: Wow, This was wonderful. I had the initial idea for the lesson because I thought it was so cool how the angle measures across a straight line will add up to 180 degrees. A simple concept, but it helped a lot of things make more sense when I recognized it. I think a lot of times we have this subconscious knowledge that we utilize everyday but don’t fully recognize. Once I started planning out how I would eventually get to a measuring activity, ideas just lead into one another, making this a lot longer lesson that I intended. There is no way I would get through the discovery part in 15 min! For me, this order of events seemed to make the concept clear. Perhaps it should be a day and a half sort of lesson?




Do you have any feedback for Jill or I about the lesson?  What would you try?


Photo credits: Jill Beauchamp, quinn.anya and bjornmeansbear @ Flickr

I had the good fortune to win a bet recently (well, best 2 out of 3) by the performance of the Yukon Huskies (jk) in the 2011 NCAA Division I men's Basketball Tournament.  My prize? A guestpost from Dave Coffey, @delta_dc.  (I was actually rooting for Butler, but that's a quality consolation prize!) This is Dave with Juneau.  Juneau is asking, "How could you bet on bulldogs?  Have I taught you nothing?  Haw!"

A few weeks back, one of our teacher assistants said, “You just started on Twitter this semester. I never would have guessed.” I wasn’t sure if she was talking about my quantity or quality. I chose quality and explained that it could be traced to Cambourne’s Conditions of Learning (a Foundational Framework of our Teacher Assisting Seminar).

This reminded me that John, my co-teacher, had asked me about blogging about the Conditions. I turned to him and said, “I’m thinking about writing about how the Conditions of Learning helped me to communicate using Twitter.” I thought this would be a good example of authentic learning in action.

The teacher assistant chimed in, “Maybe you could describe each condition in a Tweet.” John laughed, understanding that she had issued me a challenge without knowing it. Well, “challenge” accepted…


[Note from John - t was very tempting to put this in twitter-typical reverse order... but that would make it less readable.  Please forgive the lack of verisimilitude.]







Thanks, Dave and Jim Calhoun! And Brian Cambourne, of course.  The Reading Teacher has put the article introducing the Conditions online for download.  Or you can read the whole story in his book The Whole Story.  Also, I put the date on the cartoon at '95, the date of the RT article, but 1988 would be more accurate.

Photo Credit: Kathy Coffey, Rosaura Ochoa @ Flickr

The Man Who Counted is an enchanting math story book.  Originally published in 1949 in Portuguese (O Homem que Calculava), it was presented as a translation of a 13th century (Islamic calendar?, 1942 CE, maybe?) work by Malba Tahan, who is typically listed as the author, translated by Breno de Alencar Bianco.  Both of whom are fictitious.  The real author seems to be Júlio César de Mello e Souza from Brazil.

A student in Calgary gave me first copy, correctly deducing that I would LOVE it.  You can find a pdf of the entire thing online, and the copyright is complicated enough that I can't figure out if it's legal.  There's a current publication of the book, too, and it's a nice one to have.

All this is by way of introduction, as one of our promising preservice teachers, Cassie Becker, wrote up some very nice problems from the book, and was willing to share them here.  She did this for a choice workshop, where students have freedom to follow up or pursue an item of interest.  In general, I find that the students make amazing choices.

(Chapters 1-9)

Beasts of Burden

Close to an old half abandoned inn, we saw three men arguing heatedly beside herd of
camel. Amid the shouts and insults the men gestured wildly in fierce debate and we could
hear their angry cries:
“It cannot be!”
“That is robbery!”
“But I do not agree!”
The intelligent Beremiz asked them why they were quarreling.
“We are brothers,” the oldest explained, “And we received thirty-five camels as our
inheritance. According to the express wishes of my father half of them belong to me, one-
third to my brother Hamed, and one-ninth to Harim, the youngest. Nevertheless we do
not know how to make the division, and whatever one of us suggests the other two
disputes. Of the solutions tried so far, none have been acceptable. If half of 35 is 17.5 if
neither one-third nor one-ninth of this amount is a precise-number, then how can we
make the division?"
“Very simple,” said the Man Who Counted. “I promise to make the division fairly, but
let me add to the inheritance of 35 camels this splendid beast that brought us here at such
an opportune moment.”

Can you explain why this would be a good idea?

Food for Thought

Three days later, we were approaching the ruins of a small village called Sippar when
we found sprawled on the ground a poor traveler, his clothes in rags and he apparently
badly hurt. His condition was pitiful. We went to the aid of the unfortunate man, and he
later told us the story of his misfortune.

His name was Salem Nasair and he was one of the richest merchants in Baghdad. On
the way back from Basra a few days before bound for el-Hillah, his large caravan had
been attacked and looted by a band of Persian desert nomads, and almost everyone had
perished at their hands. He, the head, managed to escape miraculously hiding in the sand
among the bodies of his slaves.

When he had finished his tale of woe, he asked us in a trembling voice, “Do you by
some chance have anything to eat? I am dying of hunger.”
“I have three loaves of bread.” I answered.
“I have five,” said the Man Who Counted.
“Very well,” answered the sheik. “I beg you to share those loaves with me. Let me
make an equitable arrangement. I promise to pay for the bread with eight pieces of gold,
when I get to Baghdad.”
…
Then Salem Nazair said to us, “I take leave of you my friends. I wish however to
thank you once more for your help and, as promised, to repay your generosity.” Turning
to the Man Who Counted, he said, “Here are rive gold pieces for your.
To my great surprise, the Man Who Counted made a respectful objection. “Forgive
me, O Sheik! Such a division, although apparently simple, is not mathematically correct.
Since I gave five loaves, I should receive seven coins. My friend, who supplied three
loaves, should receive only one.”
“In the name of Muhammad!” exclaimed the vizier, showing a lively interest. “How
can this stranger justify such an absurd division?”

Each piece of bread was divided into three portions and each man ate an equal portion of bread. Can you justify this division?

The Four Fours

“Did you notice that this shop is called The Four Fours. This is a coincidence of unusual importance.”
“A coincidence? Why?”
“The name of this business recalls one of the wonders of calculus: using four fours, we
can get any number whatsoever.”

Can you make 2,5,17,26,34,91,135, etc using only four fours?

dweekly @ Flickr

Going to Market

And the shopkeeper told the following “Once I lent 100 dinars, 50 to a Sheikh from
Medina and another 50 to a merchant from Cairo.
“The sheik paid the debt in tour installments, in the following amounts: 20, 15, 10 and 5 that is

Paid 20 and still owed 30
Paid 15 and still owed 15
Paid 10 and still owed 5
Paid 5 and still owed 0
Total 50              Total 50

“Notice, my friend, that the total of the payments and the total of his debt balance
were both 50.”
“The merchant from Cairo also paid the debt of 50 dinars in four installments, in the
following amounts:

Paid 20 and still owed 30
Paid 18 and still owed 12
Paid 3 and still owed 9
Paid 9 and still owed 0
Total 50            Total 51

“Note that the first total is 50—as in the previous case—while the other total is 51.
Apparently this should not have occurred. I do not know how to explain the difference of
1 in the second manner of repayment; I know that I was not cheated, as I was paid all of
the debt, but how to explain the difference between the total of 51 in the second case and
50 in the first?”

Can you explain to the shopkeeper why this happened?

Pennington @ Flickr

Seventh Heaven

The sheik addressed the three of them: “Here is the esteemed master calculator.” And.
to Beremiz he added, “Here are my three friends. They are sheep rearers from Damascus.
They are facing one of the strangest problems I have come across. It is this as payment
for a small flock of sheep they received here in Baghdad, a quantity of excellent wine, in

21 identical casks:
7 full
7 half-full
7 empty

They want to divide so that each receives the same number of casks and the same
quantity of wine. Dividing up the casks is easy—each would receive 7. The difficulty, as
I understand it is in dividing the wine without opening them, leaving them just as they
are. Now, calculator, is it possible to find a satisfactory answer to this problem?”

Can you find a solution to the problem?


Three and Thirty

“Your total bill, with your food, is 30 dinars,” was the reply. Sheik Nasair wished to
pay the bill, but the men of Damascus refused, which led to a small discussion and an
exchange of compliments, with everyone speaking at once. At last it was agreed that
Sheik Nasair, a guest, should pay nothing and that each of the others should pay 10
dinars; so 30 dinars were handed to a Sudanese slave for his master. A few moments
later, the slave returned and said, “My master says he made an error. The bill is 25 dinars,
and he has asked me to return 5 to you.”
“That man of Tripoli is most honorable,” remarked Sheik Nasair. And taking the five
coins, he handed one to each of the three men, so that two remained. After exchanging a
“lance with the men from Damascus, the sheik handed them as a reward to the Sudanese
slave who had served them food.
At that moment, the young man with the emerald rose and, looking gravely at his
friends, said. “This business of paying over the 30 dinars has left us with a serious
problem.”
“Problem? I see no problem.” replied the sheik, astonished.
“Oh yes.” said the man from Damascus. “A serious and seeming ridiculous problem.
A dinar has disappeared. Think now. Each one of us paid 9 dinars. Three times nine is 27.
Adding to these 27 the 2 that the sheik gave to the slave, we have 29 dinars. Of the 30 we
handed over to the man from Tripoli, only 29 are accounted for. Where, then, is the other
dinar? Where has it disappeared to?”

Can you explain to the men where the 30th dinar went?

Sorry I haven't been writing more. There are many things I want to get to here, but there are many things in my way.

I had a conversation with a commenter that resulted in his writing a post to share here. He is selling something, which makes me disinclined to share it; but he is a teacher sharing his connections and story, and attempts to innovate, which I like. I've received no compensation for this. Please let me know in the comments if it is an inappropriate use of the blog, to you.

Jumping Joey's Numberline


Math Facts for Kids or What’s More Important the Answer or the Process?

By Matthew G. Mandelbaum, MA, MSEd, PhD Candidate, Learning Specialist

Two middle elementary students sit together faced with the following problem: A girl has some jellybeans that she wants to share with her friends. With 3 other friends, she has 1 left over; with 5 other friends, she has 1 left over; and with 11 other friends, she has 1 left over. How many jellybeans does she have?


After reading this problem, the two students are left with choices for problem solving. Should they take a trial and error approach? Should they continue to re-read the problem over and over again, in hopes of some insight? Or should they use a tool to help them learn? Seeking to improve frustration tolerance, perseverance, and the value of process, I suggest they use a tool. They turn to JumpingJoey’s NumberLine® Multiplication and Division Book, which they have been using to learn both operations. “It’s not just for learning number facts,” one student says to the other. “Right! Let’s figure this problem out!” he replies. They now re-read the problem with purpose, hunting for clues. “4, 6, and 12 seem important,” one student says. “Because we have to add the girl and her friends. Let’s investigate them as factors.” The students turn to each of the factor’s string of multiples and analyze the collection of number facts. “Do they share a common multiple?” the first asks. “Let’s see,” flipping back and forth through the book’s pages. “Hey, each of them has 48. Look, on the four’s page, there’s 48 (4 x 12) on the six’s page there’s 48 (6 x 8) and on the 12’s page there’s 48 (12 x 4). The common number is 48!” “Cool!” says the other. “If they all share 48 and there’s 1 left over, that means there are 49 jellybeans in total!” “Let’s check: 49/12 = 4 remainder 1; 49/4 = 12 remainder 1, and 49/6 = 8 remainder 1. That is it!” “We did it; we solved the problem! That was awesome!”

They showed a great deal of pride. I asked them if they felt confident with their answer and if they liked the process of using a tool. They said they did, because it made them feel like mathematicians; they weren’t afraid and they did not give up. In this example, a challenging problem using math facts led to an enjoyment of the process, a sense of satisfaction, and a chance to have a mastery experience where the students could take on something difficult and with there own effort, and only a small amount of adult guidance, to develop increased self-efficacy for math.

This is a path towards a sound math foundation, where process leads to performance. However, I often see that in the quest for math achievement, parents can assume that getting the right answer is the most important thing about knowing math facts, when in fact a child’s decision process that leads to the answer is what should be looked at most carefully.

Memory can work a few ways. Either there is a weak association among information or a strong association. Both types can lead to the right answer when the situation is not so difficult. However, when the challenges begin to mount, having a strong association among information will yield to fewer errors than a weak association.

The key to math facts for kids is strong conceptual understanding. Parents looking to help their children grow should seek to have benefits over the long-term in addition to short-term grades on little assessments. In order to reach this goal, parents can supplement scholastic efforts with at-home tools that provide an organized number line framework to promote conceptual understanding, mathematical fluency, plus a strong foundation for learning math facts. We feel that by using a product like JumpingJoey’s NumberLine, students build a coherent mental number line, which they use flexibly to solve a range of problems. This ability is important because, as reported in the journal Psychological Science of the Association of Psychological Science, scientists found that the quality of mental number line in children and pre-adolescents strongly and positively correlates with arithmetic aptitude, math achievement-test scores, and overall math grades.

Math facts are the building blocks for arithmetic, which form the foundation for higher math throughout the grades. It is important for the child to interact with numbers and consider their meaning with deep contemplation, because new topics will be built upon these numbers to form new knowledge. The child will need to relate what he knows to what is being taught. This relationship needs to be processed on a deep level.

When looking to help support a child’s learning math facts, parents can ask questions like “How did you get that answer?” or say “Show me your thinking.” Within this process, parents need to reward their children’s effort instead of their ability so as to build mathematical competence. To support the last goal, parents can choose products like JumpingJoey’s NumberLine that help their children see themselves as capable mathematicians who are actively engaged in the learning process. Such products should be intrinsically rewarding and promote intellectual curiosity.

Arithmetic is a fascinating subject of study. Numbers possess a lot of power. The Pre-K through elementary years are formative in establishing a child’s sense of self as a learner. Throughout these years, students are met with challenges, like learning math facts. Of vital importance, is their ability to persevere despite obstacles so that they may approach tasks with a realistic sense of confidence and openness to what is new.

Einstein said, “Never regard your study as a duty, but as the enviable opportunity to learn to know the liberating influence of beauty in the realm of the spirit for your own personal joy and to the profit of the community to which your later work belongs.” Children deserve to have this orientation towards learning. As parents and educators, we owe it to them to create an environment in which they can think this way. After all, it worked for Einstein, right?

BIO:
A New York State certified educator in Childhood General and Special Education, Matthew has over 13 years of varied experience working with students of diverse ages in a range of settings from pre-kindergarten to college, in public, private, parochial, afterschool, and tutoring environments. Together with his wife Jamie Cohen, he founded PsySoEd Dynamics® LLC, a company dedicated to developing high quality educational products whose first line of JumpingJoey’s NumberLine products helps students learn math facts and concepts using a fun, multi-sensory approach. As parents of a young girl, they’re even more committed to making math for all and fostering academic success and achievement for children. You can read Matthew and Jamie's Statement of Philosophy for Teaching, Learning and Educational Product Development here.

Copyright 2011 Matthew G. Mandelbaum All rights reserved.