Mr. Pan-tastic vs. the MathMan

Day 11, July 22, 2015

Technology in the Classroom

Flipped Learning

Apparently, the Mathman, Mr. Velisa Anusic, is the winner here. Both his learning and work experience are incredibly rich.  His association and personal encounters with Dr. Marian Small, Dan Meyer, Jo Boaler and other big math personalities he knows (but of course, does not necessarily know him) are enviable.  On the other hand, Mr. Pan (short for Mr. Panem and how my students address me) is just a proud math education “newb”.

Modesty aside (but just for a while), the technology that Ve Anusic employs in class are not new to me.  I had the chance to know and apply technology in the classroom by attending Edcamps that are organized and supported by the 21^st Century Learning department in my school board.  I am very much familiar with Google Docs, Google Forms, Learning Management Systems, Flipped Classroom, creating and using videos to support learning, instant feedback and math tools.  I do not want to specify each of these apps and web tools. You can have a glimpse of these applications, live via my Grade 10-11 Math class and Cooperative Education websites. 

Now, with so much modesty, I admire how effectively, appropriately, and interestingly the MathMan makes use of these technology in our class to support our understanding and making our learning our own. Desmos, Gizmoz, Smore.com, Wolfram Alpha are some of the few “take-aways” I wanted to introduce in the classroom to facilitate if not hasten the student’s understanding and learning of math with Mr. Pan.

Equipped with the understanding of these technology, I can perhaps, improve my instructions if I can build my lessons to follow the SAMR approach recommended by Dr. Ruben Puentudura. SAMR claims to improve problem solving skills; student engagement, motivation via publication, and differentiation in the classroom.  I think SAMR is a different way to categorize used of technology for learning.  However, learning is not supposedly linear. Learning should be exponential.  Students can learn and educator can teach more effectively by continuously creating new tasks by continuously redefining the use of technology to support the creation of new knowledge and then further create new technology to further support mathematical problem solving and thinking.  Learning is not constant, it’s a variable, that can be both dependent on and independent of technology.

                                                                                                                                                                        

Kudos to the MathMan, you and the summer ABQ course are fantastic.  My suggestion for you to improve on in the course is first to adopt a masculine name and even if life is short, wear long pants.  I hope you realize that I am just kidding.  

Finally, to my sleepless bloggermates, it has been a pleasure learning with you.  Don’t listen too much to pop music.  Math education is not all about the bass (or base), it’s all about the “Balance”.  Why rise and run if you can Pi ?

Jesse Panem

Seeley is not Silly

Day 10, July 21, 2015

Basic Facts

Nix the Tricks

Gap Closing Documents

Estimation over Calculation 

The thoughts of Dr. Small, Jo Boaler and Dan Meyer on computation, I think, are parallel.  They agree on most of the fact that teachers should focus less on calculations but more on estimation.  Use small numbers and develop first student’s number sense.  Do away with worksheets and encourage a lot of “number talk”, that is doing mental math with friendly numbers without forcing algorithm.  All of them embrace the use of technology, to handle the more complex calculations and to focus on the learners’ interest that makes the lesson more engaging.

"Faster isn't Smarter"

Cathy L. Seeley has the same line of thought with the other three math educators.  She emphasizes the importance of conceptual understanding, followed by developing math skills before actually applying mathematics to problem solving.    While there is a push from Dr. Small and Jo Boaler to calculate mentally and from Dan Meyer to effectively use technology, Seeley looks for being realistic  on when to apply mental calculations, technology over the use of pen and papers or calculators in both conceptual understanding and developing math skills.  

Don't Just Fix, Mix the Tricks

Like Seeley, Tina Cardone (Nix the Tricks)  discourages the use of shortcuts, mnemonic devices and tricks. I understand the rationale of nixing and fixing math tricks.  Memorizing the formula and rules, hampers conceptual understanding.  If we teach concept more than tricks, students will retain more.  However, I still believe that there are places for these short cuts in solving practical real world problems.  To totally nix the tricks may not be right for me.  I would keep my own techniques but not necessarily hand it over to my students.  Instead, I would ask them to develop their own tricks and give it a name of their own, which in the end is still pushing conceptual development. 

Learning by Doing

Day, July 20, 2015

Thinking Tools

Algebra / Fractions

“When I listen, I hear.  When I see, I remember.  But when I do, then I understand.”

Chinese Proverb

Today, we dealt with feedbacks, thinking tools and working with visual models for fractions and algebra.  Some of the lessons I have absorbed and their implication to my practices in the classroom.

Fractions can be represented using arrays. 

Most of us are used to solve fractions using the least common denominator for addition and subtraction.  For multiplication, we directly multiply both numerators to calculate the product-numerator and do the same with the denominators.  Then, simplify result to it’s lowest term.  For division, by multiplying reciprocals.  Manipulatives are helpful to visualize how parts are related to the whole or to other parts.  Today, I realize it is easier to visualize these operations using arrays.  Fraction problems can be solved without the use of the least common denominator, cross-multiplication, and reciprocals.

If I am to teach my students on how to solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate and proportions, I will definitely deliver instructions using arrays.  And I will prefer students to learn multiplication using the Area Method.

The Area Method is a better approach to Multiplication.

The standard algorithm is perhaps the fastest method to multiply and demonstrate the distributive property of multiplication. The Area method does the same, plus the latter promotes understanding and encourage the development of mathematical thinking. 

In practice, I would foster the use of the Area or Box method because it represents a rectangle or a square.  The area of a rectangle is the simplest: “length times width”.   A visual representation of multiplication is easier to comprehend than the numeric symbols. It also supports the important ability to estimate answers.  Further, it facilitates students’ understanding on how to multiply expressions and solving equations involving polynomials and in learning the higher level mathematics.  The understanding of the Box Method also facilitates comprehension of the utility of the Algebra Tiles. 

The use of Algebra Tiles is experiential  learning.

The reason I have hated the Algebra Tiles before  is I because I do not understand how it works.  Today,  I have learned  that is a powerful thinking tool.  This manipulative can help learners to develop concepts related to integers, algebraic expressions, equations, and polynomials.

These tiles are: big blue square (+x2 tiles); big red square tiles (-x2 tiles); rectangular green (+x tiles); rectangular red (-x tiles); small yellow (+1 tiles); small red tiles (–1 tiles) are used for negative integers (Bettye C. Hall).  How they are used to simplify, to solve, to expand, to factor, to complete the square is intuitive and exciting.

The use of these thinking tools will drastically change the way I manage my math class and hopefully translate to a better mathematical thinking for my students.

Effective Assessment is Key to Student Learning

Day, July 16, 2015

Assessment FOR, AS, and OF Learning

Triangulation of Evidence

Feedback

The Brain - Jim Craigen

According to Damian Cooper (an independent education consultant, “assessment guru” to some), there are two guiding principles to assessment: Assessment must be good to students and it must be manageable and efficient for teachers.  Assessment must promote learning, and be fair and equitable, and sensitive to individual learner’s needs. 

A lot of us are confused with “for, as, and of learning” assessments.  The following is my attempt to distinguish one from the other:

	FOR	AS	OF
Purpose	Seeking and interpreting evidence for use by learners and teachers to decide where are they, where to go in their learning and what’s the best to get there	Fostering learners’ self –reflection, presenting and modelling external, structured opportunities	Evaluating how well the students are learning and reporting of results at the end of an assignment, lesson, unit or course
Tools	Teachers observations of  and conversation with students	Self- reflection and peer evaluation strategies	Student’s products.  Descriptive feedback
Nature and Use	Diagnostic and Formative assessment to determine learner’s prior knowledge and skills	Formative helps student monitor owns progress	Summative assessment is used by teacher to summarize learning at a given point in time
Timing	Before the instruction begins	On-going, while student is still gaining knowledge and practising skills	Towards the end of learning processes but not necessarily at the end of the unit or course
	Assessment	Assessment	Evaluation

We can use the analogy of coaching a football club or game.  Assessment for and as learning is the practice.  Teachers gather data to tell learners what to do next. For informs teachers, as inform students.  Assessment of learning is the game itself. It is accomplished by the teacher and use it in evaluating the performance of the learners.  In this assessment, teachers gives the grade with descriptive feedback.  Assessment of learning is usually in the end, but not necessarily. Just like in a  football game, where winning is achieved by effective coaching, in education, effective assessment is key to student learning.

The following is an attempt to link or validate topics from previous class discussions to the presentation of Jim Craigen, today’s guest speaker, who talked about the Brain and Cooperative Learning. 

• ·         The brain activities discussed by Jim is parallel to our discussions on the Teenage Brain article.
• ·         Small talk greases social interaction (Carducci). à Dan Meyer urging educators to   start conversations and teach math through these conversations.
• ·         Emotion dictates how we are going to learn.  Positive emotions can facilitate learning.  à With open questions, everybody is loved in a math questions.  Open questions are multi entry and multi route.  That means everybody is welcome to answer questions in a welcoming and safe manner.
• ·         The more you laugh the better your memory.  à Sing and dance in your classroom.
• ·         There is 25% retention when student see instructions and 90% if they teach them.  à Open questions and parallel tasks allowing students to building problems taking ownership of their learning.  Injecting ambiguity to allow more questions and discussions.
• ·         Environmental complexity is stimulating to the brain.  à inject ambiguity in building math problems
• ·         Engagement is essential to [student] success.  à  Engagement is one of the purposes of open questions and rich tasks. 
• ·         Sensory Memory stays 1 to 2 seconds; working memory, 18 seconds.  They are to be “rehearsed” to extend them to long term memory.  à Our practice of writing blogs is a way to “rehearsing” our memory, and implementing and teaching what we learn from this course lead to long term memories and learning.

Forward Learning with Backward Design

Day , July 15, 2015

Rich Task

Today we discuss and undertake activities in Backward Design, Rich Tasks, and Problem Based Learning.  To illustrate the way I view the relationships of these topics I made up the following diagram:

        Backward design, for me, is an effective approach to planning lessons.  It means starting with the end in in mind.  Answering first the question “where are we going?” before working on the resources we need to get there.  Therefore, I start with the big idea which, in my experience as a beginning teacher, is  drawn from the curriculum overall and specific expectations.  Related to the expectations and big idea are the specific learning objectives.  Then, involving the students, I design the success criteria, which becomes the object of the assessment strategies.  All of these should be aligned to the big idea as I prepare the unit plan and the daily lessons.  Rich Tasks and Problem – Based Learning  provide an environment for teachers to develop active learners that are independent assertive constructors of their own understanding who challenge and reflect (Jennifer Piggot).

       Some of impressions and ideas and concept I understood from today’s discussions and activities:

• ·         Dan Meyer on backward design: we need to start the conversations to bring mathematics to those conversations
• ·         Dan Meyer: inject ambiguity in questions to evoke curiosity and encourage more questions and discussions
• ·         Dr. Marian Small:  in problem solving, encourage a lot of conversations.  Create divergence rather than convergence.  Contextual problems are routines.  Look for spin-offs.  Encourage creativity.  Promote arts first before the “math”.
• ·         Jo Boaler: Encourage students to work on the problem first, before teaching the “math”.  More on thinking and less calculations. There are available technologies to handle computations.
• ·         Ve Anusic on edutainment: bring in fun and creativity to stretch students’ attention span.  Sing and dance either during the class or while preparing for the class.

Communication and Consolidation

Day 6, July 14, 2015

Communication in Mathematics

Consolidation in Student Learning

Communication, according to the “Growing Success” document, is the conveying of meaning through various forms.  Through the teacher’s intervention, students are expected to express and organize their ideas and mathematical thinking using oral, visual and written forms.  They should be able to communicate for different audiences and purposes while using proper math conventions, vocabulary and terminology.

As the lead learner, I have to model communication skills specifically in preparing my class (Dr. Cathy Krpan). This is accomplished by actively and sincerely listening to my students, clearly articulating and sharing my thinking; posing effective and provoking questions during presentations; including everyone in the process; disagreeing in a positive manner; and providing talk prompts that will facilitate our discussions.

From the article discussions, I have learned that effective questions rouse student thinking and deepening conceptual understanding (Capacity Building Series).  To inculcate conceptual understanding among students,   I find the 8 tips for asking effective questions most useful and relevant: anticipate the different ways they will approach the problem; questions should be aligned to the curriculum expectations; pose open questions, those questions actually that requires an answer and those that is inclusive; incorporate verbs that elicit higher levels of Bloom’s taxonomy; and provide ample time for student to process their thinking. Further, if I have to stimulate mathematical thinking on my student, I need to guide them on how they should share and present their work, on how they verbalize their reflection and how to make connections to other ideas.

Just some comments on consolidation, since I can hardly see its connection to communication:  it is the last section of the 3-part lesson.  It is in this part, where students re-organize their thinking and continue reflecting.  This is where I see the utility of mathematical instructional strategies like Basho, math congress and gallery walk.  My role during consolidation, is to ask effective question to help student summarize their mathematical ideas embedded in the class solution.  In the end, they get engaged in metacognition and connect their own generalization to the mathematical learning goals.

Re-thinking my Mathematical Thinking

Day 5, July 13, 2015

Making Thinking Visible

Today’s discourse and activities on mathematical thinking, mathematical processes and the 3-part lesson made me realize that the way I handle my math class needs a serious and drastic makeover.   In my classroom the symptoms for reading math wrongly were evidently present:

• ·         Student do not do homework.  They do not study in advance and do not watch assigned videos.  The lack initiative even during class discussions.
• ·         Lessons that were supposedly learned in the previous units need to be repeated because of lack of retention.
• ·         Student easily give up, when faced with challenging questions.  Some may not even attempt to solve problems or answer questions.
• ·         Most of them are intimated by word-problems.  They are too reluctant to understand word problems and would settle to look for what is simple and the obvious information. When they get frustrated they ask the formula.  Then accused me of not “teaching” when they are urged to investigate the formula from the given information. 

As the lead-learner, I cannot put the blame on them.  Given the opportunity to work on another long term assignment (or a permanent position) to teach math, I will design my lessons to integrate the concepts of big idea, open questions, mathematical thinking; to plan using the 3-part lesson students; to develop patient problem solvers and independent thinkers through the practice of the mathematical processes.

Adopting Dan Meyers’ recommendation, my lesson in the future will incorporate the use of multimedia and take advantage of costless apps available in the web.  From the article on Teenage Brain, we know that technology is the adolescent learners’ expertise. Therefore, I’ll make use of their strength to divert their computer ability into mathematical thinking skills.

 From the conversation on open questions, I will allow students to learn more than what I know, evoke curiosity and encourage student intuition.  Ask short question to allow probing and investigation and force student to think.  I will let student formulate and build problems and to be less helpful and eventually moderate, if not totally eradicate “impatient irresolution” in my math class.

The Big Idea is Small's Idea

Day 4,  July 9, 2015

Open Questions, Parallel Tasks

Engaged and competent students and developing independent thinkers is a commitment to our students and to our profession.  We continuously look for ways and means to achieve these goals.  The activities and assignments we did on open questions and parallel tasks partly fulfill my quest for strategies to help me develop thinkers in my classroom.  The discussions help me re-asses my belief on how student learn:  that students should work on the same problem at the same time, and that each math question has have a single answer, (an answer that should match with what I know).

Now, I know why my long term assignment in an inner-city school went wrong.  Not too many months ago, I have had to manage math classes in Grade 11 Applied, Grade 11 Essentials and Grade 10 Applied.  For each of these course, my daily routine was to explain the lessons, ask questions, do some examples and prepare students for the weekly quizzes and the unit tests.  The results are frustrating for both the students and myself.  Differentiation is always what I thought I could have done to help my students.  However, the only strategies I manage to implement are differentiating students’ product and individually tutoring those who are struggling. 

A prior knowledge and application on open questions and parallel tasks could have helped me achieve student success in those courses.  Today, I have learned that open questions:

• ·         Reinforce a wide range of math concepts.  They require more thinking, less computational and students don’t need to memorize shortcut tricks.
• ·         Provide choice, one of the elements implicit in differentiated instruction.  They have multiple –entries and students can answer in a way that’s is suitable for their level.
• ·         Everyone benefits from different perspective when they hear other students respond, then everybody gets loved in math class.
• ·         Challenging and yet enriching and accessible to all.
• ·         Allow students to learn “how to think” rather than “what to think” and to eventually become independent thinkers.

Open questions and parallel tasks will definitely be part of my differentiating instruction and my three-part lessons.  One of the requirements of differentiating instruction is to consider the needs of each student at his or her current stage of development.  Open questions provide those choice that urges student to think and defend his or he answer rather than just finding that one right answer.  Open questions will also be a better “minds-on” activity or a diagnostic tool to understand the learner’s prior knowledge.  I can also incorporate open questions and parallel tasks into my lessons as assessment “as” learning strategy.  

If I my intention is to develop independent learners in my classroom, I should foster Dr.  Marian Small’s idea of allowing students to think differently than I do, to provide them  different choices on how to answer questions in a manner that is suitable to their level of development.  If I can offer more meaningful activities through open questions and parallel tasks, then I will have engaged, confident, competent students who will likewise enjoy learning math.

Big Idea, Smart Idea?

Today, we tackle the “Big Idea” which is deemed essential to understanding various mathematical concept as a coherent whole.  We were assigned two major tasks: First, to generate Big Ideas from the curriculum document across grade levels. Second, to formulate Mathematical questions that will satisfy, if not all, at least three overall expectations for each strand for a particular grade level.  Both approaches has advantages and disadvantages.

From the exercise, with the Big Idea approach we seem to develop goals easily, pinpoint good questions, and at the same time differentiate within the actual mathematical content.  However, we experience difficulties in establishing coherence among different strands across grade levels.  On the other hand, the second method is more straightforward since we are used to simply follow the requirements of the curriculum documents.

After the article discussions, I realized I have misconceptions on the implementation of the big Ideas.  I thought the big idea statement is something educators generate from the overall expectations.  From this big ideas we specify the learning objectives and then draw the success criteria. I also realized that this concept may not be a standalone and maybe integrated to our norm of simply selecting specific expectations in preparing our lessons.  If properly incorporated in our lessons, the big idea concept promises a better understanding of mathematics for both educators and learners.

Despite the apparent benefits of the big idea, I still have reservations to embrace this strategy.  Preparation and planning may tend to be challenging considering the amount of responsibilities teachers need to accomplish in a rigid timeframe.  A big idea? Yes, but is it “smart “ too?

I cannot do it........yet

The following are my understanding of what transpired in our classroom  on July 7, 2015:

The Biology of Risk Taking

There is so much complexity on how biological changes influences teenagers’ behaviour and how their brain works. It is the responsibility of the educators to understand these changes and the resulting behaviours that leaners experience in their puberty. Studies and developments in Psychology and Neuroscience offer some explanations and may serve as bases for provisions for realistic expectations and effective interventions in education.

“The Biology of Risk Taking” (Lisa F. Rise, 2005) recommends some strategies that educators can implement to promote healthy adolescent growth. These include understanding puberty, mentorship, long-term and continuous follow-ups, prioritizing concerns,  directing "adolescent passion" towards positive and productive ends, and collaborating with the learning community to solve problems.

As educators, when working with adolescent learners, we should keep in mind that, one, puberty is not the same for all teens.  Some of them enters that stage earlier and for others, a little bit later.  Each of these individuals experience different biological drives.  Two, they are not adults.  We should respect how they behave and think and adjust our expectations to provide them more effective interventions to promote healthy adulthood.

Growth Mindset

Carol Dweck, author of Mindset: The New Psychology of Success (2006) asserts that individuals can be categorized into their attitude towards their ability to be successful.  Some people believe that their success is a product of their hardwork and learning, while others perceive their talents and intelligence as inherent traits.  Respectively, these are “growth” and “fixed” theories of intelligence as defined by Dweck in her book.

Individuals with Growth mindset believe that intelligence can be developed and can lead to a desire to learn and therefore they are open to challenges, persevere despite of hindrances, see effort as essential to mastery, learn from criticism, and are inspired by the other people’s success. On the other hand, “fixed” individuals believes intelligence is static.  Fixed mindset leads to desire to look smart and therefore a tendency to avoid challenges, yield to obstacles, see effort as fruitless, ignore useful feedback, and threatened by others’ success.

Implications

In the classroom, developing the growth mindset can be realized by recognizing the “adolescent passion” and convert their thrill-seeking behavior into productive ends; praising “effort” more than “intelligence” and encouraging students to  learn from mistakes and failures.  In Mathematics, the development of growth mindset in students will reduce, if not eliminate, the anxiety and fear, students generally have for the subject.  Finally, I go for the recommendations of Eduardo Bricero, “listen to your (our) “fixed” mindset.  If you (we) hear, I can’t do it, you (we) add the word “yet”.