Text Set 2: Trigonometry

Text Set 2: Trigonometry

·      S.O.S. Mathematics: Trigonometry

http://www.sosmath.com/trig/trig.html

Grade level: 12.0

While the reading level of this collection of pages may be a bit high, it has excellent diagrams that present trigonometry in a variety of ways.  Here, the links between the geometric and the algebraic conceptions of trigonometry are made clear.  This is a good extension for a high performing student, or a student who has been accelerated and is studying independently.

·      Math is Fun: Introduction to Trigonometry

http://www.mathsisfun.com/algebra/trigonometry.html

Grade level:  6.8

This page from Math is Fun is written in a simple, conversational style and is accessible to both those reading below grade level and those whose math vocabulary may be weaker.  However, I find that the greatest value is in the interactive flash elements halfway down the page – students can move a point around the unit circle to create a right triangle and examine its trigonometric ratios.

·      Khan Academy:  Basic Trigonometry

https://www.khanacademy.org/math/trigonometry/basic-trigonometry

Grade level:  12.0

Khan Academy is a great resource for those students that prefer auditory methods of taking in information (most of the information is conveyed graphically or verbally).  The videos show trigonometric concepts and how to use them, in the form of notes that the author puts on a slide.  The author talks the students through what he is doing, and is clear and direct; as well, his mouse/pointer is always visible and follows which element of the notes at which he happens to be looking at any given point.  And of course, this lesson includes the interactive quiz questions that Khan Academy is famous for. 

·      Trig Cheat Sheet

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

Grade level: 2.4

This is a sheet of trig formulas and trig identities, conveniently condensed, with figures illustrating the general case.  There isn’t much reading (verbal) in this resource, but math-specific literacy is necessary for comprehension.

·      Cliff’s Quick Review Trigonometry

https://play.google.com/store/books/details?id=qo8GBrUfGIcC&source=productsearch&utm_source=HA_Desktop_US&utm_medium=SEM&utm_campaign=PLA&pcampaignid=MKTAD0930BO1

Grade level:  9.4

Cliff’s Quick Review is exactly what is says – a quick crash-course review of trigonometry concepts.  It is heavily verbal, though the reading level appears to be accessible for the majority of high school sophomores.  It is available as an e-book download from this link in the Google Play store (which is great for tech schools, though Apple users may run into some compatibility issues), as well as a hard copy from your local bookstore. 

·      Trigonometry (Corral, M.)

http://www.mecmath.net/trig/trigbook.pdf

Grade level:  11.0

Corral’s text on trigonometry is a good extension for students who would like to read trigonometry in the context of how mathematics looks when it is published within the field.  It connects trigonometry to history and other branches of mathematics.  

·      Wikibooks: Trigonometry

http://en.wikibooks.org/wiki/Trigonometry

Grade level:  7.8

Wikibooks has a good collection of information, free and available on the web to anyone, on trigonometry.  It gives us three books, organized by increasing difficulty and frequency of use of concepts present in other branches of mathematics (matrices, calculus, computing).  Topics are arranged by page, and include many diagrams.  This resource is also useful for students and teachers to think and talk about what makes a good figure – there are editors’ notes on how figures will be updated to be clearer and more useful.

·      Applications of Trigonometry (Clark University)

http://www.clarku.edu/~djoyce/trig/apps.html

Grade level: 10.9

This page gives a quick run-down of some trig applications, and the historical context in which they developed. 

·      Math Central: Applications of Trigonometry

http://mathcentral.uregina.ca/RR/database/RR.09.01/trigexamples1.html

Grade level: 8.2

This page gives a couple of unconventional examples of applying trig.  In addition, they’re examples that the average layperson can encounter, not just examples found in physics or engineering contexts. 

·      Fascinating Facts of Mathematics

http://malini-math.blogspot.com/2011/08/applications-of-trigonometry-in-real.html

Grade level: 8.6

A quick succinct list of traditional applications of trigonometry, complete with example pictures.  The pictures have figures overlaid that connect the trigonometric model with the real world context.  

·      Slideshare: Real World Uses of Trigonometry

http://www.slideshare.net/ihatetheses/real-world-application-of-trigonometry

Grade level: 12.0

The reading level on this is a bit high, but this slideshow is an excellent resource for connecting trigonometry to the real world.  It showcases how certain careers make frequent use of trigonometry, and then give details about those careers such as salary and areas of study needed. 

·      WISC-Online: Practical Trigonometry

http://www.wisc-online.com/Objects/ViewObject.aspx?ID=MSR3603

Grade level: 4.6

This is an interactive slideshow that poses some really great application problems within the context of a machine/prototyping shop.

·      Clark University: History of Trigonometry Online

http://www.clarku.edu/~djoyce/ma105/trighist.html

Grade level:  12.0

This is a brief online outline of the beginnings of trigonometry, complete with figures and historical context.

·      Applications; Web-Based Calculus

http://ualr.edu/lasmoller/trig.html

Grade-level: 12.0

This page gives a brief history of trigonometry, with a focus on its journey through various cultures and its (mis-)translations.

·      TI-Nspire: Sinusoidal Modeling

http://education.ti.com/en/timathnspired/us/detail?id=469A5FBBB74F4031882FEC9806D32867&t=3270A55B10BB4BF29C7D8883E3990203

Grade level: 6.9

This exploratory activity extends knowledge of trigonometry from geometry to algebra and statistics, and models the use of trig functions to model oscillating phenomena. 

Text Set Topic Redux

Trigonometry

Reflection 10: Technology

I think the strategies in BBR Chapter 10 were useful as a macro-structure for web-based searches, but I feel like some of the other strategies that are nested in these structures bear some investigation as well. How do students decide that a website is credible?  The -AND (Analyze and Note Details) and -SD- (Slow Down) steps from the SAND and ISSDaT strategies require students to evaluate the credibility of web sources, but do't mention more than making sure to stay away from .com sites and sites clearly trying to sell you something.  I can think of plenty of examples of .org websites that may be of dubious credibility -- I'm sure the KKK and crazy Doomsday/Rapture Preppers have websites ending in things other than .com, but they're hardly credible sources (except as primary sources to examine how bleeding insane they are).  I'm at a one-to-one school and the students almost exclusively use the web for resources when doing projects, but few of them know how to differentiate an academically credible source (e.g. a scholarly one) from one that appeals to sensationalism, trends, and popular culture -- beyond "Don't use Wikipedia."  I think a more useful strategy to analyze would be how to determine credibility, and how credibility requirements change depending on the project.

Web Resource 2: Khan Academy's Geometry section

https://www.khanacademy.org/math/geometry

Khan Academy's geometry section has been a fantastic resource for me because of the interactivity of the 'quiz' sections.  I feel like the quiz sections are almost more valuable than the videos; the videos are nice because they offer a clear verbal walkthrough of problems, but the interactive quiz sections let students experiment with figures to see if constructing counterexamples (per say) is even physically possible with the given conditions.  I feel like nothing is quite as instructive as a hands-on session of trial and error.

As well, students can sign up for an account through the website and earn points through a cute little game system and earn rewards.  In addition to the points and badges that accounts keep track of for students, student accounts can also be linked to a teacher account.  The teacher/coach account that is associated with it can view statistics for each student, including frequency of use and success rates.

One drawback to Khan Academy is that it requires student access to one-to-one technology to use successfully (in my opinion).  Even if you don't have one-to-one access, you might be able to make a couple of computers with a Khan Academy activity set up one center in a set of stations that the whole class participates in.

Reflection 9: CCSSI... everyone's favorite!

I liked reading the article that showed us that there is some difference of opinion on the CCSSI, or, more accurately, that there is some difference of opinion on how much preparation the transition will take.  I thought it was interesting to note that the article says there is some push back against CCSS adoptions, but that usually this isn't coming from teachers and parents, i.e. the educational community.  Teachers and parents seem to be largely in support of Common Core, though many think they may need some more time to develop their new curriculums and resources.  Detractors seem to be from camps that are concerned about 'looking bad' -- since CCSS will (ideally) reduce ratings inflation among schools (not every school can be 'above average'!).

One of the most interesting things mentioned in the article is some speculation on how students will react to this transition and how they will initially perform.  There is some fear that students will have difficulty with the new, rigorous, in-depth approaches to material presented under CCSS after having been acclimated to the drill-and-kill approach to material under NCLB.  I  can understand this fear, as I see it in my students every day.  The transition has not been kind to these children; being constantly asked 'why?' and 'explain' is totally foreign to these kids.  They're used to being able to pop out a numerical answer and then move on with their lives.  It's a shame that a there will be whole grade levels of students who are thrown under the bus b y this transition, because I think that in the end, after we're fully ensconced in CC, it'll be work it.

Text Set #1: Triangle Congruence

Books/Print Resources

·      Geometry Workbook For Dummies

Mark Ryan

Grade level: 12 (for teachers to use in instruction)

This book has a good ‘plain language’ approach to geometry, though the reading level is a bit high.  I would certainly use it sparingly, and mostly on concepts that don’t have much in the way of an access point for students.  Some of the exercises are also presented well and broken down into small steps and would be appropriate for students who get overwhelmed by multi-step problems. 

·      “The History and Concept of Mathematical Proof”

Steven G. Krantz

Accessed from http://www.math.wustl.edu/~sk/eolss.pdf

Grade level: 8.6

This text might be nice for advancing students who perform highly and need activities that deepen their understanding of geometry, but section 4 on the history of geometry would be a good assignment for the whole class. It could even be a read-along for those classes that struggle with reading. 

Activities/Websites

·      Triangle Congruency By SSS and Properties of Isosceles Triangles Activity

http://education.ti.com/en/us/activity/detail?id=486E4EB82B77484BBC4C0097DF3FE3E1

Grade level: 6.6

This is a great in-depth application lesson on using triangle congruency and the properties of isosceles triangles.  The lesson includes an authentic problem about building a doghouse, and set ups for analysis based on symbolic, graphic, and verbal representations.  I find this activity useful because of the depth and variety of representations in analysis. 

·      Conditions That Prove Congruency

http://education.ti.com/en/us/activity/detail?id=359A3010A7CE45C89995C27ABEC063CB

Grade level: 7.7 and above

 This activity combines conceptual and abstract verbal representations with interactive portions that let students determine empirically the conditions necessary to prove congruence in triangles.  It lets students see first-hand how changing various parts of a triangle affects congruence. 

·      Math for Morons Like Us

http://library.thinkquest.org/20991/geo/ctri.html

Grade level: 10.1

This page is a good reference for the five triangle congruence theorems, and provides great examples of small, manageable proofs for each of the cases.  It also has a little quiz at the end to test student knowledge.

·      Regents Prep: Practice with Proofs Involving Congruent Triangles

http://www.regentsprep.org/Regents/math/geometry/GP4/PracCongTri.htm

Grade level:  8.3

This activity gives good triangle congruence proofs for increasingly complex figures.  It uses the ‘traditional’ two-column form used in Geometry classes, and challenges students to try the proof before clicking to reveal the answer. 

·      Khan Academy: Congruent Triangles

https://www.khanacademy.org/math/geometry/congruent-triangles

Grade level: 11.9

Khan Academy’s collections on triangle congruency combine visual/symbolic representations with aural presentations.  It is good for those students who favor auditory methods and who may get ahead of themselves with verbal representations, thus losing the sequential logic of the proof.  Of course, after the videos, Khan Academy has some stellar interactive quiz questions.

·      Khan Academy: Congruence, Isosceles and Equilateral Triangles

https://www.khanacademy.org/math/geometry/congruent-triangles/isoscleles_equil/e/geometry_proofs_intro

Grade level: 8.8

See previous entry on rationale for videos; proof questions here are constructed as a fill in the blank for parts of the triangle with a drop-down box supplying the properties to be used as justification. 

·      Wikipedia: Congruence (geometry)

http://en.wikipedia.org/wiki/Congruence_(geometry)

Grade level: 12

Honestly, Wikipedia is actually usually a pretty good place for establishing a baseline for a mathematical concept.  Now, I wouldn’t have have the students read the entire page because it gets technical, but the intro and selected sections are good, as are the .gifs in the sidebar. 

·      WyzAnt Resources: Congruent Triangles

http://www.wyzant.com/resources/lessons/math/geometry/triangles/congruent_triangles

Grade: 10.7

This webpage uses a nice, approachable conversational tone to discuss congruence, primarily in a verbal mode.   Also, the figures here are understandable and clearly marked with different colors to indicate congruent parts. 

·      Regents Prep: Lesson on Theorems for Congruent Triangles

http://www.regentsprep.org/Regents/math/geometry/GP4/Ltriangles.htm

Grade level:  9.3

The first part of this page is nothing special and is a less stellar version of some of the other resources I’ve included in this set, but the last third of the page is dedicated to a discussion on why AAA and ASS are not valid as congruency postulates.

·      Math is Fun: Congruent Triangles

http://www.mathsisfun.com/geometry/triangles-congruent.html

Grade level: 5.8

This would be a good resource for students who do not read at a high school level.  It is simple introduction to congruence with simple figures – nothing overly complicated. 

·      Math Warehouse: Isosceles Triangle Theorems and Proofs

http://www.mathwarehouse.com/geometry/congruent_triangles/isosceles-triangle-theorems-proofs.php

Grade level: 6.7

This site is good to look at proofs in an incremental fashion, if it is a bit garish. 

·      Geometry History

http://www.kidsmathgamesonline.com/facts/geometry/history.html

Grade level: 12.0

Despite having a Flech-Kinkaid level of 12, this page provides a few basic historical facts about Geometry and geometric constructions.  I’m not entirely convinced that the reading level is as listed – I think the Greek root words discussed skew the measure.

·      Math Open Reference: Introduction to Constructions

http://www.mathopenref.com/constructions.html

Grade level: 7.9

This site gives a little bit of background on construction proofs and provides a nice answer for that student who always says “Well why didn’t they just measure it?”  It also has a collection of links to pages on constructions and proofs.

Reflection 8: Wait, I Thought We Were Finished With Vocabulary?

Nope.  It never stops.  Which was one of the things that I found interesting about the Bromely article -- she acknowledged the fact that language is a living, changing thing and that new words are created every day in English, especially in STEM fields.  I also think it's significant that she took the time and space to talk about how our attitudes as teachers towards new vocabulary affect how students perceive the experience of encountering new vocabulary.  Sheer enthusiasm for language is the bedrock that all of these other strategies are built on.  I don't know about you, but I love a good $10 word; and that makes the experience of guiding students through vocabulary instruction or discovery much less like a chore.

In general, I found the Bromely article very interesting, even if some of the statistics made me raise my eyebrows a bit.  Seventy percent of the most common words having multiple meanings seems a bit iffy to me; that just seems high.  As well, I thought that saying that English is a simple and consistent language relative to other languages was a bit of an overstatement.  While I can concede that English would be simpler than tonal languages just on a literary level, that doesn't necessarily mean that we're consistent by any means.  English pulls grammatically and linguistically from so many places it would be difficult to remain internally consistent.  In addition, I really wish the Bromely article had gone more in depth about using connections and associations to learn new words.  I know this is the primary method I use to learn new words, and by its very nature this strategy is accessible to all.  Seeing how to implement this in a classroom would be a great resource for me.

In the Baumann and Graves article, I feel like it's worth mentioning that it's really important to me that they classified symbolic representations as their own category of vocabulary.  Understanding these and knowing how to read them are crucial for building comprehension in math and science.  I often liken math to a foreign language -- it's not English, but it is a language with vocabulary and a grammatical structure, and can be understood and used to communicate.  It is internally consistent and logical.  It's not magic or gibberish.

Text Set Topic

Triangle Congruency

Reflection 7: Oops, Vocabulary Again

The BBR Chapter 5 had some strategies that transfer really well to the content area of math, finally!  Math is all about content-specific vocabulary and 'common' vocabulary that has a specific meaning in the context of a mathematics classroom.  

I really like the Contextual Redefinition strategy because it explicitly acknowledges the difference between vocabulary as seen in 'common' circumstances and as seen in academic mathematics.  In addition (zing!), I find that it is the strategy that employs the most 'think-aloud' elements.  I think that think-aloud strategies are really important in mathematics because oftentimes people view a math problem as one giant insurmountable whole written in what might as well be a foreign language.  They need to see that good mathematicians read things in chunks and frequently call up definitions and characteristics of elements of a problem as they read it.  

I also really like the Etymologia and Morphologia strategies because they add a narrative element to understanding new vocabulary and they draw in the human element through history.  However, I think I might combine the two into one assignment/strategy in my classroom.  Why did the person who named this particular thing choose the words or morphemes that they did? 

Feature Analysis also struck me as being particularly useful in mathematics because we are constantly comparing concepts or things that share many characteristics but differ in a few minute but crucial ways.  Feature Analysis charts are a nice way of keeping track of the nitty gritty that sometimes gets lost in the concise mathematical definitions.  

Reflection 6: Help for Struggling Readers

The Daniels and Zemelman chapter on strategies for helping struggling readers helped me realize where some of the bumps in one of my classes may be coming from.  The idea of forming mental pictures and scouring word problems for pertinent data is something I should take into account when I assign word problems; I've noticed my students have trouble going from a purely verbal description to a diagram, a crucial skill in geometry.  I also liked that one of the main strategies that they focused on was that of having readers make note(s) of where they become confused.  This helps students develop self-monitoring skills as well as be aware of content area knowledge they may need extra help with.  I like that they mentioned that the teachers in the vignettes had set up a chain of actions that were to be taken at the points where students become confused, but I wish that they had explored those actions a little more deeply.  Where should students be going and what should they be doing after they realize they've lost the thread of a reading?

Reflection 5: Vocabulary Development

I loved the Bean, Baldwin and Readence chapter on developing vocab.  It has lots of awesome strategies for getting students physically and mentally engaged with the vocabulary that's essential for understanding.  (You don't learn new words by osmosis!)

We actually did a semantic mapping activity the other day to start off Module 2 in my Geometry class to talk about what it means to 'study harder' -- things you can do by yourself, things you can do with others, and things you can do in class. (This discussion was prompted by a reflection ticket I had them do after the Module 1 test; they were to list 3 good things they're doing, 2 bad things they're doing, and 1 thing they're willing to change in the next unit to do better.)  I used a semantic mapping software for the iPad called MindMeister, found here.  It's pretty lightweight and easy to use on the fly, after you get a feel for where to tap to add new bubbles.

I also really like the idea of the feature analysis chart in general to compare and contrast things, but I think that I might use it later this module to wrap up quadrilaterals.  It'll be a nice way to recap all of the different properties, and to look at which quadrilaterals are actually more than one type (a rectangle is also a parallelogram).  I'm also thinking about having students do something like the verbal/visual representations strategy for each shape, as we learn them.

I'm stoked that this reading had so many strategies that are easily transferred to my content area!

Reflection 4: Textbooks v. Literally (ha!) Anything Else

The Daniels and Zemelman chapters this week were all about putting textbooks in their place. And you know what? I agree. Textbooks are fantastic reference books when you need some quick, basic facts or a general knowledge of a topic; but they're hardly engaging, deep explorations of important ideas. For me, the math book is where I send my students if they ask for a step-by-step algorithmic solution for a problem type or for extra practice problems to drill and kill. It's not what I'm going to use for primary instruction, especially in light of the graphic layout of our current Geometry textbook. "Browser windows" clog up the page and cutesy CGI people pop out of nowhere; it's difficult to establish a flow for reading. Speaking of flow, all the talk of 'narrative' nonfiction made me think of my History of Math textbook; there were some sections written in that style, and I might find them useful in my Geometry courses to give some history and reason to geometric theorems as constructions (as opposed to a rule to be memorized). I also had an interesting discussion after I read these chapters about how these textbook failings don't seem to apply to college texts nearly as often. Why? Because college texts (outside of gen ed courses, usually) are generally written by an expert with a focus in that particular field, and address only that specific topic rather than trying to explain a whole brand of study at once. These texts are written by people who 'discover' the knowledge, not middlemen in the exchange of information.

Reflection 3: Reading Strategies and Textbooks, Oh My!

Chapter 5 of Daniels and Zemelman was chock full o' reading strategies. I myself really liked the fact that the strategies were grouped by time of deployment; not all struggling or reluctant readers have difficulty during the same part of the reading process. I know I could definitely use some of the 'before reading' strategies with my geometry class -- the students can read the passages competently and tell me what was in them, if they bother to read the passages. But they have no motivation to do so from the outset,and thus don't read; or, when they do, we haven't discussed any prior knowledge to which students can connect this new information. It seems like the students view texts in that course as disconnected and dull. I can pick and choose strategies according to the needs of my students. Chapter 6 talks about the dreaded textbook and how we as educators can compensate for its shortcomings. One of the strategies discussed is that of essentially writing your own textbook for your course. I've had some professors who've done this, with varying degrees of success. I know my mentor teacher is doing a version of this right now, also with mixed results. While it's good to pare down and dig deeper into more essential topics of a subject, you're relying on the judgement of one individual to make the distinction between these topics and those that are superfluous. Also, this instructor may not be a stellar instructional writer, or more likely, a great layout designer. My mentor is using bits and pieces of other texts to build his own Frankentext with which to teach; and while we cover everything we need to, the formatting is often wacky and it's difficult to determine which ideas are the most important. In short, we lack text features!

Web Source Review 1: TI-nspire CAS

TI-nspire CAS is an app that mirrors the functionality of the new TI-nspire calculators. It is available at in the iTunes store for a hefty $29.99, though your school can negotiate a license so the app is free of charge to students and faculty. (Tough luck, Android crowd; but as Apple has cornered the education tech market at the moment...) The nspire app lets you do calculations on a 'scratchpad' screen, much like you would on a traditional TI-89 calculator. It's capable of detailed color graphs for functions, making spreadsheets and lists for statistical analysis, and constructing interactive geometric figures. It uses a file structure similar to that of a computer, so one can create a file with section 1 problems in it, and then a separate document for each problem that contains a calculation scratchpad, function graphs, statistical analysis, etc. to encompass all the various representations of that problem. These files can be saved, so navigating between problems doesn't mean you have to lose all your work. The app is, honestly, notoriously difficult to use at first, but with practice, it has incredible utility for fluidly navigating different representations of problems, which is something I know my students need to see and practice more often. As well, the TI website has interactive lessons and simulations that you can download. Students can do things like drag to alter a function and see how it changes the equation in real time.

Reflection 2: Content Specific Reading Skills and Text Sets

This week's readings in Tovani had us consider how exactly literacy is used in content areas besides English, and what skills are relevant in different content areas. Reading a math text is different than reading a short story is different than reading an article from a science journal, and all require specific literacy skills to be read meaningfully. The book talks at length about how we as teachers are expert readers in our content areas and fluidly and automatically use our content-specific skills to make sense of a text -- but our students are not (yet) expert readers in this area. They don't know to employ these skills that we use without recognizing it. It is crucial that we, as good readers of content-area text, slow down our thinking and strategy use so that students can see how good readers handle particular types of text. I can think of two ways this is particularly relevant to math: first, math has its own terse variety of English; second, math textbooks often have a particular structure that is sometimes difficult for students to follow. Math uses English in a very compact way, where modifiers drastically change the meaning of a noun, but yet no redundancies are built in. (Consider the example "If A and B are finite sets, |A| is greater than or equal to |B|, and f is a function f:A->B, then the function f is surjective," and "If A and B are finite sets, |A| is greater than or equal to |B|, and f is a function f:A->B, then there exists a surjective function f." The first is false, the second true; and the the key difference is "the" or "a" -- the definite versus indefinite articles.) Modeling precise reading and interpretation of mathematical properties is a crucial aspect of my pedagogy because it is not a skill present in other disciplines. I would have to explicitly discuss the meanings of 'the' and 'a' in a literary sense in the previous example, and talk about the mathematical implications thereof. Math also requires a particular heuristic for problem solving. I had a discussion recently with some other teachers in my school's math department about how much I hated KWL charts (sorry Sterg), which sparked a conversation about using them in other subject areas or as framework for a lesson. As it turns out, we found a modified KWL chart online that is perfect for scaffolding word problems for students. (I'm sure I'm not the only one who's seen the panic-give up response to word problems in students.) The structure aspect of math texts is particularly interesting for me at this moment because in one of the classes I'm co-teaching, we use presentations made from various book and website clippings for direct instruction. Students are allowed to use these presentations as resources on their mini-quizzes, but I've had several quizzes turned in blank because students can't seem to find the relevant material in their presentation notes. Now that I've considered how important structure (even just on a graphic/layout level) is to the understanding of a mathematical text, this may be attributable to inconsistent structures among our sources and a lack of strategies in my students for recognizing different structures in math text.

Reflection 1: The 30 Million Word Gap, SREs, and the Value of Culture

The Hart and Risley article in essence says that children in higher SES environments have experience with around 30 million more words than students from families on welfare by age 4. This gap in experience continues to widen as children age, and is predictive of the achievement of students. To me, its worth noting the constraints of the study; as it was a longitudinal study, they needed families that had long-term residence in the area and families that did not mind having researchers come in to observe household interactions every month. This automatically excludes transient families and families where various illegal activities may be taking place (who clearly won’t want to be observed) -- families who arguably offer less support for exploratory and linguistic experiences for their children. This means that the gap may be larger than that found in the Hart and Risley study, and that the ratio of positive to negative comments made to children may also be skewed. The Fitzgerald and Graves article explains how scaffolded reading experiences can help, in particular, English language learners and how some of these experiences can be implemented. The goal here is that English language learners don't have to sacrifice content-area learning in exchange for reading skills, nor are they sacrificing learning English skills for content mastery in their native language. The article also talks about how language is a medium for cultural understanding; that learning a new language and syntax means also learning a new cultural outlook. In a twisted sense, one could almost look at families in poverty having their own distinctive culture and language, as evidenced by the Hart and Risley study; and thus we come perilously close to a discussion about the relative worth of cultures in these two articles. I think it would also be interesting to see if the type of gap seen here would be found again in a sister study of children in Mexico, France, China -- anywhere else with a public school system.