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Thursday, June 2, 2011

Diagnosing Teachers’ Needs

As an Instruction Math Coach, I am considering the themes of math instruction with my cohort of teachers.

  • Content knowledge
    • Understanding network of connections
  • Disposition towards mathematics
    • Personal feelings about math
  • Pedagogical content knowledge
    • Developing and maintaining learning communities
    • Knowing how to “unpack” big math ideas
  • Underlying beliefs about learning
    • Teaching styles
    • Learning styles
  • Interpreting student thinking
    • What do students know and how do they show it?
    • What are misconceptions, partial knowledge and common confusions
  • Assessing prior knowledge
    • What might we assume students already know and how can they access this?
  • Habits of planning
    • Lesson planning and long term planning: when and how detailed
  • Engagement with curriculum material
    • What is role of core curriculum vs. standards based instruction?

Daily Thought

With the possible exception of the equator, everything begins somewhere.  

C.S. Lewis

Math Coach's Observation Protocol

Today I begin a new phase as Math Instructional Coach.

This is an observation protocol I am working on:

What is the evidence that important mathematics is at the core of the lesson?
    • Teacher has clear summary of lesson
    • Teacher can respond to unexpected student questions
    • Discern between important math concepts and tangential ones
What is the nature of the interactions between teacher and students and amongst the students themselves?
    • Confident
    • Respectful
    • Who is talking the most?
    • Teacher asks probing questions
    • Time to think
How are visual aids and models used to facilitate student understanding?
    • Whiteboard/overhead/ELMO
    • Sketches/diagrams
    • Posters
How does the grouping strategy address individual student needs?
    • Whole group, individual, partners, small groups
    • Which children work together
Does the lesson pose an appropriate range of challenge for every student?
    • Accessibility of the problems?
    • Extensions?
How is the management style conducive to develop a learning community?
    • Behavior management offers opportunities for student reflection?
    • Behavior expectations and consequences are explicit?
    • Teacher decisions leads to student autonomy?
How does room arrangement and placement of supplies help the learning goals?
    • Conducive to student collaboration
    • Private spaces for students to work independently
    • Neat and organized classroom
    • Student access to supplies and manipulatives

Friday, May 27, 2011

Thursday, May 26, 2011

No End of the Year Class "Party" for us

My students have all been clamoring for a "party".

They expect me to buy pizza, they bring chips and soda. Classroom parties are just big food fests.

So many of my students are really quite overweight. If I were to buy them pizza, they would eat it in class and still go to lunch and eat the horrendous food there.

I will NOT be buying pizza, nor allowing the chips and sodas in my room.

I think we will play games. Active Games.


Cost of Iraq War: How many seconds of war = teacher's salary

Digging around I find that there are no hard, cold facts readily available about the cost of the Iraq war. Depending on political bent and accounting practices, the numbers range from the billions to more than 3 trillion.

I found this table, which broke down the numbers in an interesting fashion:

It got me thinking that the average teacher's salary would be covered in less than 20 seconds. Wow!

California and the Common Core Standards

I was looking around the Internet to see the status of Common Core Standards in California. I found the excerpt below, indicating that they have been adopted but have run against a moratorium of funding for new framework initiatives. Part of me really understands this moratorium because it has felt that we have been buffeted around with new initiatives ever 3-4 years. I am not convinced we need new materials right away, at least not for my school district, which uses Everyday Math, but I do believe we need to look at what we have and reorganize the scope and sequence to reflect the focus points at each grade level.
Good teaching should never be based solely on the materials, but on the standards being expected of the students.

The standards approved by the SBE are based on the K-12 English-Language Arts (ELA) and math standards created by the Common Core State Standards Initiative and augmented by California's Academic Content Standards Commission to establish learning objectives with the same level of rigor embodied in California's existing standards. Those existing standards will remain in effect while state officials consider when and how to implement the new ones.
Cost is a major factor affecting the implementation timeline. California will need to invest in new curriculum frameworks, which guide standards-based instruction, plus the development, adoption, and purchase of new instructional materials. In addition teachers and school leaders will need training, and the state testing system will need to be aligned with the standards.
Those steps will be significantly delayed unless California lawmakers lift a moratorium on updating curriculum frameworks and adopting new instructional materials enacted in 2009 and lasting through 2013-14. Even if the moratorium is lifted, the State Board of Education may not adopt instructional materials in math until November 2014 and in ELA until November 2016, according to a proposal that the CDE presented to the board in November 2010.

What are Common Core Standards and why do we need them?

What are educational standards?
Educational standards help teachers ensure their students have the skills and knowledge they need to be successful by providing clear goals for student learning.
Why do we need educational standards?
We need standards to ensure that all students, no matter where they live, are prepared for success in postsecondary education and the workforce. Common standards will help ensure that students are receiving a high quality education consistently, from school to school and state to state. Common standards will provide a greater opportunity to share experiences and best practices within and across states that will improve our ability to best serve the needs of students.

Standards do not tell teachers how to teach, but they do help teachers figure out the knowledge and skills their students should have so that teachers can build the best lessons and environments for their classrooms. Standards also help students and parents by setting clear and realistic goals for success. Standards are a first step – a key building block – in providing our young people with a high-quality education that will prepare them for success in college and work. Of course, standards are not the only thing that is needed for our children’s success, but they provide an accessible roadmap for our teachers, parents, and students.

Wednesday, May 25, 2011

Common Core Standards in Math at the 5th Grade

There are three main pillars upon with 5th grade math learning rests:

1. Fractions
2. Division
3. 3 dimensional shapes and their measurement

Here is what the Common Core Mathematics Standards have to say:

Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

Daily Thought

A teacher is a compass that activates the magnets of curiosity, knowledge, and wisdom in the pupils.

Plausible SCOPE and SEQUENCE: 4th Grade TERC math curriculum

I did a little research about TERC (I used to teach from the older version, but it is rather similar as the new one).

I would offer this as a suggestion for changing up the scope and sequence:

1. Shape of Data
2. Factors, Multiple and Arrays
3. Landmark Numbers
4. Fraction Cards and Decimal squares (supplement with fractions circles and more mixed number stuff, but I like the connections between fractions and decimals)
5. Multiple Towers and Division Stories
6. Size, Shape and Symmetry (particularly area and perimeter and angles)
7. How many packages and how many groups (2 digit multiplication)

If there is time, do the Penny Jar unit (but that could also be interspersed throughout the year in smaller activities for a day or two)

Skip the 3-D unit (it is not in the 4th grade Common Core Standards and tends to be just outside the developmental range of many 4th graders)

That is 7 units in the year. I would include some small pre and post test for each unit to see how students have grown and to show parents what is being learned in class. 

Sept: Data
Oct: Factors
Nov/Dec: Landmark #'s
Jan/Feb: Fractions
Feb/March: Multiples
April: 2-D Geometry
May: Packages and Groups

I would also make most of the homework relate to multiplication, division and fractions, and less about addition and subtraction. That would send a different, more rigorous message home than the current sequence seems to be doing.

Teacher Memo

We all should be encouraging respect for teachers. 
They know more than students about a great deal of things.
They try REALLY hard to share what they know and how they know it.
They respect effort.
Let's respect their effort as well.

How dare you NOT use fraction manipulatives in 4th grade?

As the fourth grade math year runs to it end, consider this:

Were fractions at least a quarter to a third of the math you covered during the year?

Did you know that fractions: equivalence, like denominators and mixed numbers are one of 3 focus standards in the new Common Core Standards adopted by many (almost all) of the 50 states in the US?

A clear understanding and broad base of experience with fractions are crucial (CRUCIAL) for a positive and useful algebra experience later on (and yes, we 4th grade teachers are partially responsible for the experience down the road).

It is not just "another topic" to cover: it is one of the three pillars of fourth grade mathematics.

Fractions taught in isolation without manipulatives is useless: so don't try cramming it all in numerically. Fractions is a HUGELY important CONCEPT, not simply another SKILL to check off.

Why should you use manipulatives?

Manipulatives serve four purposes:
  1. They engage the senses. Multi-sensory tools have been shown to dramatically increase understanding and retention. Many students, especially those who had difficulty in school, need multisensory tools to learn effectively.
  2. They help students discover concepts. This is especially true for math concepts. It can be difficult, for example, to explain how 2/4 = 1⁄2, but with when students physically moves shapes, they can prove it for themselves.
  3. They help keep students focused. Everyone appreciates a little variety. By varying activities, you help your student to stay focused and keep learning.
  4. They encourage practice. Students will often practice more with manipulatives than they will with worksheets.

Daily Thought

Best is good. Better is best.

Tuesday, May 24, 2011

A College Degree Pays but Major Important for Making Money - ABC News

A College Degree Pays but Major Important for Making Money - ABC News

That English major might seem more fun than math or engineering, but it won't pay off nearly as well in the long run, according to a Georgetown University study released today. Researchers compared the average earnings for various fields over a 40-year career and found a 300 percent difference between the average salary of the highest-earning petroleum engineers and the lowest-earning high school guidance counselors.

The classroom I am leaving behind in 2.5 days.


Our school ends on the Friday before Memorial Day Weekend. We started in the second week of August, so our 176 days of school (4 furlough days discounted) are about to finish up.

This has been the best classroom I have ever had the opportunity to occupy. I will miss it, but more importantly, will miss the kids who populated it. I loved teaching ancient history, but starting next year I will be a PRIME Math Coach. 

I love history, love reading, love writing.

But it is math that makes my heart beat fastest!

Math Anxiety: Real or Imagined (and does it matter if it is one or the other?)

In his foreword to Skemp’s The Psychology of Learning Mathematics, Foss stated that
mathematics is a curious subject, psychologically. It seems to divide people into two camps…there are those who can do mathematics and there are those who cannot, or who think they cannot, and who "block" at the first drop of a symbol. (cited in Skemp, 1971, p.9)

Math Anxiety is a topic that interests me. I am not 100% convinced it is a true anxiety. I am also not a psychologist qualified to say what is and what is not an anxiety. The term "math anxiety" is often used lightly and quickly adopted by people who find adversity in the math classroom (and outside it as well). What is not clear to me is whether this is a domain specific anxiety. For example, could there not be a reading anxiety, a science anxiety, or a writing anxiety? Is there really anxiety in play, or is there some sort of learned helplessness and negative feedback loops?

These are on my thoughts as I investigate math anxiety and the elementary teacher for my new math coaching job next year.

The Math (and use?) of Linkedin?

I am a newbie to Linkedin.

In fact, I didn't want to join yet another social network (same connections, new passwords = ugh!)

But it seems that it might be a good idea to join, set up a professional network as I look to spread out my wings a little and try towards new horizons.

So I looked around and saw this info graphic ( I like to work these things out via NUMBERS ).


Still I am not convinced of the role that Linkedin has for education.

If you know, please clue me in.

Along the way: connect with me via twitter: pepepacha


Do the Math: Eliminate Tax Cuts Not Equal to Eliminate Jobs

I found this article very illuminating. It takes mathematics and applies it directly to a political issue. Rather than fanning the flames of partisanship, it cooly looks at the issue of taxes from a more pragmatic point of view. That is another thing I love about math!

My wife and I run a small lodging business in a resort town in New Mexico, employing eight people. Income from that enterprise approaches the bracket that would be directly affected by elimination of the Bush tax cuts. 

We have pondered the personal ramifications of this issue, and have come to the conclusion that increasing our taxes on earnings of more than $250,000 from the current 35 percent to 39.6 percent would have virtually no effect on our spending habits or our lifestyle. More importantly, it would not affect our ability to hire another employee.

Here are some real numbers to illustrate my point.

We typically start employees at $12 per hour. There are 2,080 hours in a working year. Multiplying those two figures gives the cost of one employee for one year: $24,960. Add in FICA and Medicare and we're up to $28,277 per year. Let's remember this $28,000 figure. 

Now, let's say that business picks up and our taxable income increases by $100,000. Leaving the Bush tax cuts in place, we would have $65,000 to spend as we see fit. Repealing the Bush era tax cuts for the wealthy would leave us with $60,400. 

If we really needed an additional employee because of added work demand, that $4,600 difference would just about fall off the priority scale in the decision to hire someone or not.

If business demanded the additional employee, we would hire them and happily go home with the remaining $32,000. The tax rate increase of 4.6 percent doesn't and wouldn't affect our decision to add an employee. This slight increase simply doesn't add up as the job killer the Republican leadership says it is!

Finally, and most importantly, the Republican leadership correctly points out that a large portion of the taxpayers in the over-$250,000 bracket is actually small businesses, much like our own. This is true, but if a small business is struggling so much that it is in danger of folding, it obviously will not even be in this higher income bracket and would not be affected by repealing the Bush tax cuts for the wealthy. 

With that thought in mind, it is clear that allowing the Bush tax cuts to continue for those taxpayers in the under-$250,000 bracket, and letting those cuts expire for those of us above that figure, would actually preserve some of these same struggling small businesses, and create jobs down the road as the economy improves. 

The country would be better served if our legislators imposed tax increases on those of us who can afford to pay them, rather than spouting political generalities that are unsubstantiated and numerically challenged. 


Daily Thought

No tree has branches so foolish as to fight amongst themselves. 

— Ojibwa Indian Saying

Monday, May 23, 2011

Teacher Memo

Thinking that good teachers are like good basketball players - Don't panic, make adjustments on the the run, and never give up.

Common Cores Standards, Fractions and the fourth grade experience

I am a big fan of the Common Core Standards. I think they bring rationality and clarity to a complex field. I love how they build upon themselves logically both within a grade level and amongst them.

Here is what the Common Core standards say in their 4th grade introduction:

In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

And here is the fraction work, broken down:

Extend understanding of fraction equivalence and ordering.

  • 4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
  • 4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

  • 4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
    • Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
    • Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
    • Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
    • Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
  • 4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
    • Understand a fraction a/b as a multiple of 1/bFor example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
    • Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
    • Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Understand decimal notation for fractions, and compare decimal fractions.

  • 4.NF.5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
  • 4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
  • 4.NF.7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

1 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.
2 Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.

Teach Less, Learn More? Singapore Math

Singapore Strengths

Framework: The study indicates there is a correlation between focused frameworks such as those used in Singapore and good test performance. Singapore offers an alternative mathematics framework for lower-performing students that covers all the mathematics topics in the regular framework, but at a slower pace and with greater repetition, and with support from expert teachers.

Textbooks: Singapore’s textbooks build deep understanding of mathematical concepts while traditional U.S. textbooks rarely get beyond definitions and formulas.

Teaching: Singaporean elementary school teachers are required to demonstrate mathematics skills superior to those of their U.S. counterparts before they begin paid college training to become a teacher. They receive a high level of professional development training (100 hours) each year.

Assessment: Singapore uses more challenging tests and utilizes a value-added approach that rewards schools for individual student progress over time.

This website also has this to say about US Math Strengths (not a couple of words I see connected together very often)

U.S. Strengths: Although the U.S. mathematics program is weaker than Singapore’s in most respects, the U.S. system is stronger than Singapore’s in some areas. The U.S. frameworks give greater emphasis than Singapore’s to developing important 21st century mathematical skills such as representation, reasoning, making connections, and communication. The frameworks and textbooks also place greater emphasis on applied mathematics, including statistics and probability.

TLLM, “Teach Less, Learn More” initiative


Daily Thought

"Why is it that nobody understands me, yet everybody likes me?" 

Albert Einstein

Game of Thrones Math

Game of Thrones is a great show, thanks to its heavy reliance on character development, political intrigue, and suspense – not to mention the excellent acting from all parties involved. I will say that it beats the book for me, which is not something I say lightly. The story moves more fluidly and the characters are easier to distinguish in the show when compared to the book. I watch it on my iPad HBO GO app, so I don’t have to be a slave to scheduling. 
Game of Thrones started out with somewhat underwhelming ratings, those ratings have steadily risen with every new episode.
The ratings, courtesy of The Game of Thrones experienced a ratings high with 2.6 million people watching the 9 p.m. Combined with its encore, Thrones was up to 3.3 million viewers for the night, with the show averaging 8.1 million viewers per episode across all platforms
The premiere pulled in around 4.2 million viewers, so doing the math: 
8.1 ÷ 4.2 means approximately a 93% rise in viewership. Of course, this is based on a statistically count of all supposed “viewers” on cable. HBO can potentially know how many viewers they have via their HBO GO site, but they can't know with the same certainty the number of viewers on cable itself. They know the numbers of subscribers, but have to do additional studies to know which shows these viewers are watching.
At any rate, it is good enough statistics to get the show renewed for next year, making this Math Teacher happy.

Saturday, May 21, 2011

Pizza math

If you have a cylinder with a raduis "z" and a height of "a", then what is the volume? 

found on: