Prove Your Mathematical Thinking

Good mathematicians explain their solutions and can prove their reasoning.
It’s not enough for a student to get the correct answer. We learn through discussion, through sharing our ideas with others. When teachers ask, “How do you know?” How can you prove what you know?” It compels students to think more deeply and to justify mathematical concepts through informal deduction. Stage three along Van Hiele’s scale of Geometric thinking encourages us to look at the properties of things. Stage one: visualize. Stage two: analyze. Stage three: informal deduction. Stage 4: deduction. Stage five: Rigor. This final stage address non-Euclidian (Spherical) Geometry where all lines meet at the poles.

What do I have questions about?
Questions I have are how can we encourage quiet students to engage in dialog in sharing their solutions? In a classroom where proving solutions is the expected norm, what does a forum for ongoing dialogue look like? How can we record, track, post, and reflect on student work in a more public manner. Is there a Math wall dedicated to the days thinking and the “whole class” ideation? How can we elevate the status of low performing students and give them an opportunity to succeed in a more dynamic conversational environment?

What are the implications for classroom practice?
Asking higher-level questions and not settling for correct answers, by asking kids to prove their ideas, is a way to invite discussion and reflection into a lesson. We can ask why? How did you solve that? What does another student think of what you said? Can you Prove it? A second implication for classroom practice is collaborative work – SOLVING PROBLEMS IN TEAMS -  I believe this is an essential strategy for teaching math to kids in the classroom. It’s easy to over think group work. A way to simplify this in my mind, is to imagine playing games in teams. 2-on-2. For instance, to extend a manipulative lesson, ask kids to solve their problem in pairs. The first one with a justifiable solution earns 2 points and moves 2 places around a board game, a bonus round could include proving how they know this is true (deconstructing the problem), before moving to the next shape. And so on.

Scientists at Play!

Today, I observed a group worthy task in action while exploring the force of water on a wheel. Students were given an array of materials (manipulatives) with little instruction and asked to solve a problem. The objective was clear: lift a weight from the floor to the desk by using the force of water. Communicating explicit criteria increases the classroom interaction. Repeating the objective several time, posting on the active board, and asking the class for a choral response to demonstrate understanding, are a few ways to reinforce the message and to keep kids on task. Building on prior knowledge of mathematics vocabulary, the concept of surface area was given a practical hands-on application. During this highly engaged activity, students were lead to discover surface area as a powerful factor in helping the wheel to turn. Not only did this Science lesson incorporate mathematics, but also the structure of the group effort taught communication skills and team collaboration. A concern with group work however, is ensuring that all students are engaged through having clearly defined roles. I noticed that one person withing a group was called the “getter” (those who gather the materials). Other potential roles in the group dynamic are: facilitator, resource manager (getter), recorder/reporter, and a team captain.

Another benefit of group work is the sharing that takes place after all groups have an opportunity to problem solve. First, is the jigsaw strategy of requesting an individual to leave their own group and to observe another group’s progress. After an individual is chosen, he/she comes back to report the findings to their original group.  Second, a class may be prompted with numerous questions before, during and post lesson. How do we build a water wheel? What did you notice? What force drives this action? Is there another way to manipulate these interlocking wheels to have more surface area? While a class that is engaged in high quality discussion may be an early predictor of the average level of writing performance, Lotan (2003), excited voices, captured in the moment of learning are often eager to share. I look forward to seeing the results of the follow up writing and water wheel re-design lesson. Using the prompt, “What did you find the most challenging?” a host of hurdles were offered. Discussions of the different results between experiments, water pressure vs. water weight were mentioned. (two different types of force) Higher order thinking compare/contrast statements can happen in real time. Students begin to question why some solutions perform differently than others. Importantly, they begin to infer and to see what to try next. How can we make this process easier next time? How can we improve our designs? “We poured the water instead of spraying it and that made it spin really fast.” Perhaps the greatest benefit of group work is the sharing of group ideas. It may be a difficult task to reflect and substantiate ones thinking, yet, having a willingness to critique and participate in this part of the process can lead to improved individual and group learning, Lotan (2003).  Shaping an environment for quality discussion, presenting clear criteria and encouraging a willingness to critique and substantiate ideas leaves only one question in my scientific notebook. Are group worthy tasks worthy of teaching? I have seen students theorize, explain, and become immersed in continuous investigation who would probably answer, “Yes!” The funny thing is, I don’t think they would call it learning. On this day, they are simply scientists at play.

The day all animals could talk.

“The day all animals could talk,” is an engaging topic for 4th grade writers to explore. One student, I will use the pseudonym Allan, explored this fictional, narrative prompt with vigor...a scarlet macaw parrot hides away in a blanket, surviving a snowy trip to school. This story is playful, imaginative and fairly well organized.

Beginning with classification of Ideas, in the six traits of writing, Allan’s demonstrates relevant and telling quality details that move beyond the obvious and the predictable (NWREL, 1992). Allan admitted that his weakness is having too many ideas and wanting to write about everything. He goes back and crosses out or his teacher will delete a whole block of copy. I paraphrased by saying, “So, it sounds like you need help focusing your story?” Adding lots of detail is good, but only if it supports the one thing the story is about— the topic. Allan’s story builds to enrich the central theme. The parrot in his story,  played games such as: “Pin the tail on me!” and “trampoline.” He tells the story with humor, “Imagine having 24 parrots jump on you. It’s humiliating.” His organization sounds inviting when snow is falling like “meteors.” A satisfying close ties back to the beginning of the story in the parrot’s own words, “It’s freezing out there!” Compelling information carries the reader throughout. Allan’s writing style is entertaining and dynamic offering solid word choice such as, “making a ruckus.” His fluency has an easy flow, rhythm and cadence, “In the morning, all was quiet. I have to admit…” Sentence structure varies in length form 5-15 words, with about 66% close to the 15 range. Voice is captured with sensitivity in the subtle details of how the parrot spoke about his life in being separated from his family. Conventions are overall very reasonable, with infrequent miss-spellings, idol to (idle), and with numerous examples of self-correction during the editing and publishing phase.

When I asked where his ideas come from, Allan responded, “I write and write and write, and then I go back and cross out what I don’t want.” Clearly, he has ownership of the writing process and sees himself as a writer. Allan exhibits an eager attitude and a firm understanding of the process of using graphic organizers. Regularly using a 1st , 2nd and 3rd writing draft he takes out and adds on content. Allan is also interested in tetherball, soccer and ping-pong. His scientific mind embraces future potential writing topics of liquid, solid and gas, water vapor and salinity, and the life cycle of insects.
Spelling development for this student reveals a need for working with doubling, rippen (ripen); suffixes, sivilies (civilize); and the initial consonant sound of a hard “c” in kattle (cattle); and oppisition (opposition). A discussion of the base root (civil) welcomed, “Oh, like civil with a ‘c’.” After sharing a host of base words beginning with (oppose), and surrounding the topic of  table tennis (opponent), Allan quickly understood the meaning of this new word (opposition).

Concluding thoughts for this student are: While topic sentences are clear and concise, I suggest improving his organization by developing an original title.  I suggest teaching with greater emphasis on thoughtful transitions and guiding towards the expression of personal voice. Finally, to improve vocabulary, spelling lessons may surround the areas of suffixes and base roots.


Lesson Plan:
"SUPER SUFFIX COMIC BOOK" 

Objective(s)/Learning Target(s):
Students will learn to improve their writing through creating a based word suffix chart.
Students will apply their learning of suffixes by creating an eight-panel comic book.

Standard(s):
GLE 1.3.1:  Revises text, including changing words, sentences, paragraphs, and ideas
GLE 3.1.1: Analyzes ideas, selects a narrow topic, and elaborates using specific details and/or examples

Instructional Strategies / Teacher Instruction:
This small group lesson will occur in two stages.

STAGE 1  :20
I will discuss how suffixes affect word meaning.
I will distribute directions with 20 base-root words and 8 suffixes.
I will model creating a chart with 21 rows and 9 columns. Suffixes will be listed horizontally across the top (-ed, -er, -est, -ful, -ing, -ive, -ly, and -s/es).
Base words will be written and fall vertically down the page in the left column.
I will ask students to consider and cross check the dictionary to be sure they are recording REAL words.
Color in the empty boxes with marker.
Share your chart with the class. Compare/contrast results.

STAGE 2  :20
Students will create their own cartoons using dramatic illustrations, dialogue bubbles and words with suffixes from the previous lesson.
I will ask the students to divide their oak tag into 8 sections. (one section per scene).
I will encourage students to create a new story line on a personal topic of interest.     (perhaps giving super-powers to an inanimate object)
I will remind students to use a smattering of base words with suffixes from their chart.
I will look for discussion points on word and passage meaning as affected by suffix choice.

Materials/Preparation needed: 
A large piece of oak tag, a ruler, a pencil, a dictionary, a set of directions, colorful markers, 11 X17 paper. Chart answer key.

Suffixes: -ed, -er, -est, -ful, -ing, -ive, -ly, -s/es
Base Words: act, blame , bold, bubble, burn, clam , cheer, color, create, drive, elect, happy, help, jump, last, like , pass, play, sick

Assessment of Learning:
I will note to the degree students are able to accurately complete their suffix chart.
I will project an answer key on the active board and ask students to self assess.
I will look for learning opportunities by comparing inconsistent class chart results.
I will request a gallery walk, giving writers the opportunity to share their cartoons.
I will request that each student present or speak to at least one base word and one suffix in their final product.

ScienceHouse

iTouch mini apps are infiltrating the airwaves. Science can be brought into the classroom with short videos that are designed to inspire and excite kids of all ages. With topics such as: Chemistry, Convection and Fire, the ScienceHouse app is free and definitely hot! Science Class Experiments are brought to you by Science House and feature Science Teacher Dan Menelly, winner of the NSF Einstein Fellowship in Cyberinfrastructure!

The limitless applications that are available today, usher in a new era in teaching that can motivate and inspire while instruction is differentiated in the classroom. Whether fast finishers have the opportunity to watch an extended science lesson, or those who are more visual learners have an immediate means of seeing and hearing the instruction for a 2nd and even for a 3rd time, video is certainly a compelling way to teach. While choices seem endless, however, limitations do exist. First, not all students can afford the technology. If the school supplies the equipment, it will need to be managed with firewalls and instruction. In fact, apps have a new rating system all their own. As a word of caution, this rating system should be closely reviewed and ranges from 4+, 9+, 12+ and 17+.

While high energy students and short attention spans may be engaged by the always on, fast-paced appeal of video game-like iTouch apps, I wonder if teaching more moments of silence and controlled breathing can help students to regain a better sense of focus and concentration. The upside, is that relevant content can now be delivered immediately, repeatedly, and in a way that is accessible at all times. The downside, is that iTouch apps have an enormous power to engage and at the same time are at risk of being addictive. My hope is for educators to have the foresight to teach students to be responsible with this technology. For more... educational apps click here.

Algebraic Reasoning: Concrete to Abstract

Today, while practicing teaching best practices of negative integers, I learned (1) the importance of teaching numerous strategies for solving math problems, and (2) the effectiveness of moving from concrete to the abstract when teaching algebraic reasoning.  What is subtracting, after all, but adding the opposite.  To quote a colleague, “Negative numbers doesn’t mean it doesn’t exist. It forces the value in an opposite direction.” An abstract thought indeed. In the real world, we may relate this to the debt side of an accounting spreadsheet, or a host of engineering and science career paths. Yet, how do we teach? We begin with the concrete method of manipulating black and white beans. We point to and count aloud along a number line, or walk facing forward and then walk backward along a masking tape line on the floor. We can list the pattern in a series of related numeric equations, so kids can “see” the answer in a new way. We can manipulate poker chips; base ten blocks, rods, cubes and sticks. We can flip over algebraic tiles from positive to negative as we move from concrete demonstration and practice, to the lattice method, and finally to the abstract “Foil” method (first, outer, inner, last). Using multiple strategies can help students of different learning dispositions to solve, factor and determine the product of two binomials.

Getting kids to open up and explain their thinking is one way to welcome a host of solutions into the classroom. “What do you think about __________’s idea?” or “Next?” This encourages reflection and conversation which is student-centered. Reinhart, S. (2000) shares, “I concluded that if my students were to ever really learn…they would have to do the explaining, and I the listening.” The implications are that kids not only get it, but also thrive to become teachers themselves. The result is more guided teaching in the class, less direct instruction and more demonstration of student knowledge.

The uncomfortable struggle all students face, is showing confidence in the face of the unknown. Will the teacher call on me? Will I open my mouth and stick my foot squarely in it? Can we set the environment for a safe caring classroom making it okay to say, “I’m stuck.” I plan to offer excessive wait time in my classroom to gather more thoughtful response. Seeming like an eternity, I will move on and offer, “(student) I know you have valuable ideas to contribute, will you promise me that you will raise your hand on another question when you have something to share?” It is important to validate the effort. Struggle makes our brains grow!

Numeric and Geometric Patterns

Today, I learned that group worthy tasks have multiple entry points for every problem. When students solve math problems together, share their ideas, and when multiple solutions from the group are valued, learning takes place at an accelerated rate. Math is no longer obsessed with finding only the right answer but by finding a host of solutions. Indeed, as human beings we naturally see patterns in the world. In the “Sneaky Snake” math problem below, I stumbled upon a realization. First, I saw a geometric pattern where a central block increased by one tall and by one wide. I wrote a numeric equation for each consecutive group. When comparing this new group, I saw yet another pattern emerge. 5 = (6 x 4) + 2,   6 = (7 x 5) + 2,   7 = (8 x 6) + 2.  I could see that the next number in line would be one greater and one less than the number “n”, where 2 is constant. This lead to the algebraic equation of n = (n - 1)(n + 1) + 2.  The remaining question I have is how many ways can the formula be solved?  I welcome pattern recognition in a linear numeric manner and in a geometric visual approach, and I can see the benefits of encouraging both in the classroom. While there is always a definitive right or wrong answer, there is never a single solution. How we get there is up for grabs. It all depends on how we look at it.