Curriculum Contemplations
Your One-Stop Shop for ELA & Math Happenings for September
Student Agency via a Growth Mindset
As we work through this space of increasing student agency, it becomes imperative for us as educators to understand how mindsets can impact not only how students see themselves as learners, but how our own personal biases and beliefs impact interactions with students, colleagues, and curriculum.
The phrase Growth Mindset has been bandied around quite a bit the past year or two in the world of education. The positive social connotation describes the belief that various talents and abilities are developed through motivation, determination, and understanding. Being comfortable with taking risks, reflecting on failures, and accepting challenges are some of the attributes associated with those deemed to have a growth mindset.
This is contrasted with a Fixed Mindset, which is the belief that talents and abilities are unevenly rationed if you will, or reside in genes that are passed on to future generations. It has been suggested that those of a fixed mindset persuasion limit their opportunities for success. Arguably even more detrimental are the occasions when a fixed mindset impacts others' commitment to learning.
As with most constructs, misconceptions conflate the underlying truths.
Having a growth mindset is not:
- to be confused with effort. "Certainly effort is key for students' achievement, but it's not the only thing. Students need to try new strategies and seek input from others when they're stuck. They need this repertoire of approaches - not just sheer effort - to learn and improve" (Dweck, 2015).
- simply providing students with generic praise. Praise that is specific to student learning and that which highlights the learning process (including the setbacks), helps students develop an appreciation of the challenges that often accompany deep understanding.
- an excuse as to why students are not progressing in their learning (i.e. "The student has a fixed mindset").
- all or nothing. As educators, it is important to acknowledge that in certain instances or situations, we might exhibit characteristics more aligned with a fixed mindset. Facing that reality head-on, rather than disavowing its existence, helps us purposefully transform said actions to be more in alignment with a growth mindset.
This journey actually begins with us as educators, perceiving ourselves as learners and analyzing our own mindsets, before we can hope to foster a growth mindset in our students. "Growth-minded educators collaborate with colleagues and work to strengthen their practice. They believe that students can learn and succeed, and don't blame them for failure, instead teaching them to persist and learn from their failure. Rather than saying 'These kids can't learn. They won't even try,' educators with a growth mindset say, "How are my assumptions about these kids getting in the way of their learning? What do I need to change in order to reach them?' (AVID Professional Learning, 2016).
As human beings, we are not always able to control which mindset drives our thoughts, actions, and words, but we can choose how we react to our mindset. When we react with a judger (i.e. fixed) mindset, the mind focuses on placing blame and distinguishing the winners from the losers in a situation. If we substitute growth for a learner mindset, as suggested by Dr. Marilee Adams, we shift our attention to thoughtfulness, supported by questioning to seek solutions. This helps to develop a win-win situation.
So, as we encourage our students to develop a growth mindset, which will lead to advanced levels of student agency, we will want to keep a finger on the pulse of our own mindset, and mindfully react to whatever our current state of being might be. Through our reflective actions and improved self-awareness, instruction will continue to flourish. Thank you for all you do to provide the Sunnyside student community with an excellent education.
ELA Elaborations
English Orthography
English orthography is the alphabetic spelling system used by the English language. English orthography uses a set of rules that governs how speech is represented in writing. In English, this definition also includes its grapheme-phoneme (letter-sound) correspondences.
A grapheme is the written representation (a letter or cluster of letters) of one sound. There are 26 graphemes, or letters of the alphabet.
A phoneme is the smallest unit of speech sound. The English language contains approximately 44 phonemes (depending on dialect). The 26 letters of the Latin alphabet (individually or in combination) represent the 44 sounds in English. To show that we mean a sound, not the grapheme, we place the phoneme between slashes, called virgules (/). The sounds in the word chop are represented as /ch/ /o/ /p/.
Phonemes can be divided into two major categories – vowel and consonant phonemes. A vowel sound is made when the airstream has a continuous flow. Although there are only 5 vowels in the alphabet, they make up 19 vowel phonemes:
5 Long Vowel sounds
5 Short Vowel sounds
3 Dipthongs (/au/, /ou/, /oi/)
A long and short oo (2 sounds, Long /oo/ as in spoon and Short /oo/ as in book)
4 ‘r’ controlled vowel sounds (/or/, /ar/, /er/, /air/)
Of course, there are many ways to spell the 19 vowel phonemes.
A consonant sound is one in which the airflow is obstructed when a sound is made. The airflow can be cut off, either partially or completely, by the lips, tongue or teeth. There are 21 letters that make up the remaining 25 phonemes.
What do we mean by text complexity?
Text Complexity is...
“The inherent difficulty of reading and comprehending a text combined with consideration of reader and task variables; in the Standards, a three-part assessment of text difficulty that pairs qualitative and quantitative measures with reader-task considerations.”
CCSS Appendix A
Although there isn’t an exact science for determining the complexity of a text, teachers can use their professional judgment and consider three factors when determining a text for students to read.
A three-part model is used in Common Core for measuring text complexity.
1 Qualitative Measures
The qualitative measures of text complexity require an informed judgment on the difficulty by considering a range of factors. The Standards use purpose or levels of meaning, structure, language conventionality and clarity and the knowledge demands as measures of text difficulty. (pg 6, CCSS Appendix A)
2 Quantitative Measures
Quantitative measures of text complexity use factors such as sentence and word length and frequency of unfamiliar words to calculate the difficulty of the text and assign a single measure (grade level equivalent, number, Lexile etc). There are many formulas for calculating text difficulty and, while they provide a guide, the readability or difficulty level of a text can vary depending on which formulas or measures are used. (pg 8, CCSS Appendix A)
3 Reader and Task
The third measure looks at what the student brings to the text and the tasks assigned. Teachers need to use their knowledge of their students and the texts to match texts to particular students and tasks. (pg 9, CCSS Appendix A)
Mathematical Musings
Mathematical Strategies vs. Mathematical Representations (Tools)
What’s the difference between a math TOOL and a math STRATEGY?
Over the past 5 years of implementation of the common core standards, teachers have been inundated with new ideas, vocabulary, expectations, and requirements. Two things that primary teachers embraced early on were the use of math manipulatives and encouraging students to talk about math. After all, this wasn’t such a stretch; Unifix cubes, double sided counters, and bingo chips had been in use forever! It was an easy transition to incorporate questions such as “How do you know?”, “Why did you choose that tool?”, and “What was your strategy?”. However, in implementation to meet our students’ mathematical needs, it seems that a math tool has been generalized as a math strategy and vice versa.
When we ask Sarah what strategy she used to solve today’s word problem and she replies, “I used a ten frame”, many are pleased that she a) remembered what a ten frame is called, and b) that she came up with the right answer. But, is what she said correct? Is a ten frame a strategy? Or, is a ten frame a tool?
Math Strategies
A math strategy is a way of thinking about math. Students may need to use a tool, or not. A single tool might be used with several strategies, and a single strategy might be used with several different tools.
Math Tools
A math tool is anything that helps a mathematician to keep track of their thinking. Pencils, paper, counters, drawings, rods and units, cubes, dice, chips, rulers, calculators, sticky notes, fingers, an eraser, crayons, pebbles, sea shells, ten frames…..all of these can be used as tools in math. If you can count it, sort it, stack it, or use it to label, or organize; it can be a math tool.
Math Strategies typically used for addition and subtraction:
Direct Modeling: Count All—All sets represented, all steps modeled, and the action is followed exactly. Students might use fingers, counters, drawings, base ten blocks, ten frames, etc.
Counting: Count On and Count Back—One set is represented when solving problems
Count On from First-the student counts forward from the first addend in the problem
Count On from Larger-the student first identifies the larger of the two addends and counts forward from there.
*Both counting strategies require a method to keep track of the number of counts that represent the second addend. Students might use fingers, counters, or tally marks. However, many children develop a mental strategy and give no indication of a physical action to go with their counting.
Derived Facts—Student uses ten facts, or make a ten, doubles, doubles plus or minus one (or another fact they may have from memory)
Grouping—Numbers are decomposed by place value or grouped by friendly/landmark numbers
Student uses place value to combine or take away numbers
Recall of Number Facts- Student utilizes memorized facts
Incremental –Counting on by 10s
Compensation-Rounding a number up or down
Student Invention-Student consistently uses an unusual strategy that always yields an accurate answer.
Adapted from: Children’s Mathematics: Cognitively Guided Instruction
Making Sense of Problems
Of the 32 students interviewed, 75% of them gave numerical responses. Here are some observations from the 32 students:
- 2 students calculated the answer to be 130 (125 + 5)
- 2 students calculated the answer to be 120 (125 – 5)
- 12 students calculated the answer to be 25 (125 ÷ 5)
- 0 students calculated the answer to be 625 (125 x 5)
- 4 students stated that they guessed their answer (90, 5, 42, and 50)
- 4 students tried to divide 125 by 5 but could not correctly implement the procedure
Three particularly interesting students included:
- The student who found the shepherd’s age by using the old “add the sheep and dogs and divide by two” trick and got 65. You didn’t know about that trick?
- The student who got 120, then said that you don’t have enough information to figure out the shepherd’s age, then seemed to feel so uncomfortable with that conclusion that the student decided to guess 90 years old.
- The last student who explained that the reason for dividing was because it didn’t say “sum,” “difference,” or “product” in the problem.
Something to think about is the idea of intellectual autonomy. Many of the students knew the question was a little crazy and yet they felt like they couldn’t question the questioner and were obligated to provide an answer. So, they repressed that feeling and gave an answer anyway. That’s certainly not where we want students to be. While it is important to be respectful of authority figures, you can be respectful and still express your thinking.
As for Math Practice 1 from the Common Core’s Standards for Mathematical Practice, here are expanded versions of the quotes from the video:
- Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
- They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
- Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
There’s no better way to see if your students are making sense of problems than to ask your students this same question. You have to make sure you keep a straight face and have them write their answers on a piece of paper. When everyone is done, have them crumple the paper into a ball and throw it at someone else in the classroom. After a few rounds of this, the papers will be sufficiently mixed that you can then ask each person to open up the paper and read what was written to have a conversation about their classmates’ answers. When you’re done with the conversation, show them this video so that they realize they’re not alone. After you try it out with your students, please shoot Maggie an email about how it went, along with the students’ grade level.
adapted from robertkaplinsky.com
Rules That Expire
1) Use Keywords to Solve Word Problems
The keyword approach is frequently introduced early in students' mathematical education as a way to solve word problems. However, using keywords encourages students to overgeneralize by cherry-picking numbers from the problem and using them to perform a computation, disregarding the context of the problem. Sense making is thus removed from the process. Many keywords are common English words that have multiple meanings. Often a list of words and corresponding operations are posted so that word problems can be translated into a symbolic, computational form. For example, students are told that if they see the phrase in all in a problem, they should add all the numbers given in the problem. Although the keyword quantity sometimes signifies the need for addition, at other times it does not. Take the following problems as examples: Tina has 3 cookies, Chris has 5, how many cookies are there in all? Compare that to Cookies come in bags of 15. If you buy 3 bags of cookies, how many cookies do you have in all? Keywords become especially troublesome as students explore multi-step word problems and must decide which keywords work with which component of the problem. Although keywords can be informative, using them in conjunction with all other words in the problem is critical to grasping the full meaning.
Expiration Date: 3rd Grade (3.OA.8)
2) Multiplication is just Repeated Addition
Considering multiplication as only repeated addition can result in students thinking that the expression 3 cubed is equivalent to 3 + 3+ 3. This thinking leads to over generalizations because students come to believe that 3 raised to the third power means that 3 is used as an addend 3 times. Writing such expressions in correct expanded form can help with this misunderstanding.
Expiration Date: 6th Grade (6.EE.1)
3) You Cannot Take a Bigger Number From a Smaller Number
Students might hear this phrase as they first learn to subtract whole numbers. When students are restricted to only the set of whole numbers, subtracting a larger number from a smaller one results in a negative number, so this rule is true. Later, when students encounter application or word problems involving contexts that include integers, students learn that this "rule" is not true for all problems. Consider the following: A grocery store manager keeps the temperature of the produce section at 4 degrees Celsius, but this is 22 degrees too hot for the frozen food section. What must the temperature be in the frozen food section?
Expiration Date: 7th Grade (7.NS.1)
4) Move the Decimal (Scientific Notation)
Students tend to get the answer correct half the time, and do not understand what went wrong the other half of the time. Students are thinking about moving the decimal place, not about place value or multiplying by 10. Students need to ask themselves, "Is this a big number or a small number?" Scientific notation gives us a compact way of writing long numbers, so is this long because it is large or because it is small? An important idea of scientific notation is that you are not changing the value of the quantity, only its appearance. Writing a number in scientific notation is similar to factoring, but in this case we are only interested in factors of ten.
5) FOIL
FOIL implies an order - precalc students are sometimes shocked to learn that OLIF works just as well as FOIL does. It is also a one trick pony. There are other ways to multiply binomials that are transferable to later work such as multiplying larger polynomials and factoring by grouping. When students memorize a rule without understanding they misapply it. See student work 4a and problem 6 for examples of this.
If we instead replace FOIL with the distributive property, it can be taught as soon as distribution is introduced. Students can start by distributing one binomial to each part of the other binomial. Then distribution is repeated on each monomial being multiplied by a binomial. As students repeat the procedure they will realize that each term in the first polynomial must be multiplied by each term in the second polynomial. This pattern not only carries through more advanced versions of this exercise, but relates to students knowledge of partial products from the elementary grades.
Expiration Date: HS Algebra 1
adapted from Cardone, 2015; Karp, Bush, & Dougherty, 2015; Karp, Bush, & Dougherty, 2014
ELL Tidbits
Throughout our teaching careers, we have experienced situations where students are reluctant to participate in the classroom, as well as situations where everyone was eager to speak up. These contrasting experiences began to raise questions, such as:
What hinders or facilitates classroom participation?
In what ways can we encourage participation from all students?
How can other students encourage the participation of others?
In what ways do our biases impact expectations for our students?
Student participation is a crucial component for the learning of all students. This is even more important for ELLs who are developing language alongside their content knowledge. In the classroom, language use positions students. These positions are not physical, but metaphorical, and represent opportunities that one has to participate and communicate (Harré & Van Langenhove, 1999). Therefore, positioning is critical to each student’s success and learning in the classroom. How students are positioned can affect their ability to develop communicative (e.g., using language to demonstrate understanding, seek information through questioning), academic (e.g., content knowledge, norms, and practices), and social (e.g., interacting with others in context-appropriate ways) competencies (Pinnow & Chval, 2014).
ELLs can often be positioned inequitably in peer-to-peer and whole-class discussions, which makes it difficult for them to gain access to academic debate and discussion. Inequitable positioning constrains ELLs’ access to learning opportunities necessary for developing both advanced content abilities and English language proficiency. Every teacher positions. It is not a matter of if one positions it, but how. Research examining classroom interactions emphasizes the role of the teacher in promoting ELL academic success and participation (DaSilva Iddings, 2005; Verplaetse, 2000; Yoon, 2008). Yoon (2008) argued, “The main reason for [ELLs’] anxiety, silence, and different positioning has much to do with being outsiders in the regular classroom context (p. 498)” (p. 23). When ELLs were positioned as powerless or troublesome, ELLs became isolated and less interactive in the classroom. Contrary to popular belief, Yoon (2008) noted that the English language proficiency levels of ELLs could not explain ELL participatory behavior across classroom contexts. Rather, when teachers practiced productive positioning of ELLs, ELLs engaged in the classroom rather than withdrawing into silence and isolation (Yoon, 2008). In this study when ELLs were positioned as powerful by the teacher this changed the participatory behavior of ELLs and the way they were treated by their native-English speaking peers. This research also shows that content teachers are not always aware of their role in positioning ELLs.
So this leads us to consider: How can ELLs get the conversational floor? How does their voice become part of the learning community? Positioning is critical to student’s success and learning in the content classroom (see figure below). Of course, this is complex and time sensitive. The beginning of the school year is a crucial time to be thinking about the positioning of our students. “Positioning starts in how we as teachers view others. If we have a deficit model perspective, a limited view of ELLs’ potential, this thinking will be present then in our words and actions towards ELLs” (Pinnow, 2014). If instead, our words and actions convey a perspective of tapping into students' limitless potential, our students participation may not negatively impact their learning.
Parent Partnerships
The Parents’ Guides to Student Success, from the National PTA, were developed by teachers, parents and education experts in response to the Common Core State Standards that more than 45 states have adopted.
Created for grades K-8 and high school English, language arts/literacy and mathematics, the guides provide clear, consistent expectations for what students should be learning at each grade in order to be prepared for college and career.
These documents can be linked to your class website, highlighted in your class newsletter, or referenced during parent-teacher conferences, in an effort to communicate and clarify our year-end educational goals for students.
Tech Tools You Can Use
Teaching & Learning Department
Tammi Baushka - Literacy Program Specialist
Rebecca Ridge - Literacy Program Specialist
Julia Lindberg - LAD Program Specialist
Kristel Foster - LAD Program Specialist
Maggie Hackett - Math Program Specialist
Donna Rishor - Math Program Specialist
Email: margaretha@susd12.org
Website: susd12.org
Location: 2238 East Ginter Road, Tucson, AZ, United States
Phone: 520-545-2000
Facebook: facebook.com/SunnysideUSD
Twitter: @sunnysideusd