Uncategorized
Sense Making
One of the reasons I am so passionate about basic math understanding is a beautiful 24 year old girl named Cameron. She was a wonderful student, but in her final years of high school she struggled with math. In a very matter of fact tone, this daughter of mine will tell you she hates math. Now, I don’t think math needs to be “one of her favorite things” (Sound of Music reference in honor of our favorite musical of all times), but I do think it would benefit her life to at least understand and feel confident that she can do math.
Last year, while Cameron was home from Chicago for the Christmas holidays, I happened to be working on a blog post about the division of fractions. (The reason this is still timely a year later is because of the two national presentations I have done recently on the subject and my work with the common core.) I decided to pose the problem in the post to her and her initial reaction was negative. Being a mom, and a teacher, I wouldn’t let it go. Being a good daughter she decided to humor me and play along.
First you need to know that Cameron was driving us down the interstate at the time, so there was no opportunity for paper and pencil calculations or the use of manipulatives. The whole “math lesson” occurred through mathematical discourse and sense making. My first question to her followed the tack of my blog post, “Banished From the Math Classroom, Don’t Ask Why….”. At first, Cameron insisted she couldn’t think of a reason to divide 3 1/2 by 3/8 but before long we arrived at 3 1/2 pizzas and 3/8 of the pizza being a serving. (For those of you who know me, I apologize. I never let my students use pizza as the example but I was working with a reluctant learner who is a self avowed math hater. I wanted her to continue and I was trying to build self confidence.) With some step-by-step logical thinking Cameron arrived at 9 servings with one piece left over. With only a little coaching on my part, she also identified the leftover piece as 1/3 of a serving.
Now, I am not deluded enough to think I converted my daughter that day, but she did say no one had ever helped her understand division of fractions in that manner before. I am sure there were a lot of other concepts that she never had a chance to “make sense of” and that has led to her “attitude” about the subject. I will continue to work on her over time but I hope that we can eradicate this “hatred” of math from our classrooms by spending more time in activities that lead to dialogue and sense making.
“Sum”thing to work on!
Deep, Not Wide
For years, teachers have bemoaned the fact that they were forced to teach a mile wide and an inch deep. The new Common Core State Standards for Mathematics are designed to reduce the number of concepts that will be taught at each grade level, thereby allowing teachers to devote more time and depth of instruction to each standard.
The concepts placed at each grade level were determined based on prerequisite topics that were to have been taught in a previous grade, international comparisons and the professional judgment of educators, researchers, and mathematicians.
These standards do not dictate a predetermined order of instruction or a particular curriculum. As it says in the Common Core document, “These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.”
Just “sum” thing to think about!
New…..and Improved? Assessments
When it comes to discussions of the common core, the conversation quickly turns to the question uppermost in teachers’ minds, “What will the new state assessment look like?” The truth of the matter is, we aren’t sure yet.
Initially, the word was that the new math assessment would be groundbreaking. It is supposed to include selected response, multiple mark, constructed response, computer simulation, and open-ended questions as well as performance based tasks. Both of the assessment consortiums, as outlined by Deb Haneke, in Assessments and Common Core Standards, are charged with helping ensure students who graduate from high school are college and career ready. Their plans for the new assessments are similar but not identical. The Smarter Balance Assessment Consortium, of which Kansas is a governing state, is planning an adaptive assessment; one in which the computer aligns the difficulty of the questions to the student’s performance level.
At this point, there seems to be some backpedaling when it comes to the news about the new assessments. According to an article in Education Week, Experts See Hurdles Ahead for Common Core Tests, there are several factors that may impede the creation of a truly innovative assessment. As in most areas of education today, the lack of money rears its ugly head. Although both consortiums received Race to the Top Assessment grant funds, this does not include long-term funding for test administration or revision. With this in mind, there is pressure to get the test right the first time. The timeline for development is also a factor that may impede the amount of innovation that we see in the final product.
So my advice to teachers at this stage of the game is to focus on the standards themselves and leave the test, for now, to the test makers.
Just “sum”thing to think about.
From Basic Math to Calculus: That Looks Easy
For those of you looking for help on everything from basic math to calculus, physics or history check out the Khan Academy website. According to the site, “The Khan Academy is an organization on a mission. We’re a not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere. All of the site’s resources are available to anyone. It doesn’t matter if you are a student, teacher, home-schooler, principal, adult returning to the classroom after 20 years, or a friendly alien just trying to get a leg up in earthly biology. The Khan Academy’s materials and resources are available to you completely free of charge.”
Sounds like a pretty great deal and from the math videos I have watched, they do a nice job of showing more than just the procedural side of math. They were also touted by a very smart and personable 14 year old author, Adora Svitak, in the final keynote at this year’s AESA conference. If the Khan Academy had anything to do with helping educate this young lady, it is worth a look.
The Khan Academy is “Sum”thing to check out and Adora is “Sum”one to check out!
A Whole New Look
For teachers who are proficient at navigating their current state standards, the Common Core Standards may have a completely different look. In the area of mathematics, the content of the document is divided into domains, clusters, and standards. Domains are the overarching term and refer to a large group of related standards. Clusters are groups of related standards. Because mathematics is a connected subject, standards from different domains and clusters may be closely related. Finally, the standards define what students should understand and be able to do. An example from fourth grade would be: Domain – Operations and Algebraic Thinking; Clusters – Use the four operations with whole numbers to solve problems (three separate standards), Gain familiarity with factors and multiples (one standard), and Generate and analyze patterns (one standard).
Within the document, each grade level does not necessarily have the same domains or number of domains as the preceding grade. However, the standards are aligned vertically from one grade to the next. Unlike some current state standards, the Common Core State Standards do not spell out every step in the instructional process and some do not have example problems. Therefore, educators are going to have to investigate the new standards thoroughly and translate them into new instructional practices.
“Sum” thing new to think about!
When Five Plus Five Equals Eight
For the last several months, I have been privileged to work with hundreds of teachers as they unpack the Common Core State Standards. In each session, which have been held in grade level groups, we have worked through the meanings of the 8 Standards for Mathematical Practice and each time one point becomes extremely obvious. These eight standards, or measures of student behavior, are inextricably linked. It is no surprise really, as they were developed using information from the National Council of Teachers of Mathematics (NCTM) Process Standards and the National Research Council’s (NRC) Strands of Mathematical Proficiency. The point is driven home, however, when teachers work through the process of identifying for themselves what each standard means and does not mean for their classroom.
NCTM’s five process standards include:
Problem Solving
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving
Reasoning and Proof
- Recognize reasoning and proof as fundamental aspects of mathematics
- Make and investigate mathematical conjectures
- Develop and evaluate mathematical arguments and proofs
- Select and use various types of reasoning and methods of proof
Communication
- Organize and consolidate their mathematical thinking through communication
- Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
- Analyze and evaluate the mathematical thinking and strategies of others;
- Use the language of mathematics to express mathematical ideas precisely.
Connections
- Recognize and use connections among mathematical ideas
- Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
- Recognize and apply mathematics in contexts outside of mathematics
Representation
- Create and use representations to organize, record, and communicate mathematical ideas
- Select, apply, and translate among mathematical representations to solve problems
- Use representations to model and interpret physical, social, and mathematical phenomena
The five stands of mathematical proficiency are:
(1) Conceptual understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students] to learn new ideas by connecting those ideas to what they already know.” A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors.
(2) Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
(3) Strategic competence is the ability to formulate, represent, and solve mathematical problems.
(4) Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.
(5) Productive disposition is the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p. 116)
When you combine the two you end up with these 8 Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
As teachers begin to discuss each of these eight standards they see how the process standards and mathematical proficiencies are interwoven throughout. Examples of what that looks like for their classrooms and their practice can be found on videos at: http://www.insidemathematics.org/index.php/common-core-standards
“Sum” thing to think about!
Tidbits from NCSM
Last week I attended a meeting of the Kansas State Mathematics Leaders. A good portion of our day was spent discussing the common core state standards and what is known about the future mathematics assessment. Several of the members had attended the National Council of Supervisors of Mathematics national convention and reported back to the group. Here are some of the things they shared.
· The new state assessment will include the opportunity to learn (OTL) component that is currently available in Kansas at the high school level. This means a 12-week testing window at the end of each school year will allow for two opportunities to take the state assessment. Schools will be allowed to re-teach and re-test students who do not reach the level of mastery expected for that grade level on the first administration.
· The reporting of assessment results will be in the form of a growth model, which may mean each student would have to be pre-tested or screened at the beginning of each school year.
· Concepts will not be retaught from one grade to the next. Mastery of grade level content is expected at the end of each year. Therefore, it is imperative we develop support structures for struggling students early. They will need access to the regular curriculum and additional support for content not mastered in a previous grade. RTI and MTSS will play a critical role in the common core standards process.
· The concerns about the common core for math that were enumerated at the convention included: a disregard for technology, due to political debates; the overload of material to be covered in sixth grade; concepts only being taught once; the need for vertical discussions about what and when to teach content; the need for professional development for teachers in both math content knowledge and pedagogical skills; and the fact that everyone must teach the common core state standards for their grade level or the system will break down.
Where Do We Begin?
A journey of a thousand miles begins with a single step. ~Lao-tzu Chinese philosopher (604 BC – 531 BC)
With every new task we undertake, there is the question of where to begin. Sometimes the first step is obvious and other times there are various options. Moving to the Common Core State Standards could be equated to the journey of a thousand miles, in that it requires new learning for the majority of teachers in the nation.
Obviously, one of the first steps is to acquaint oneself with the grade level and subjects one is required to teach. However, on a larger scale, it is important to be familiar with all the standards. It is important to understand what content students have already been exposed to, been expected to master, and what they will be learning in the future.
Another critical component to be aware of is the transition timeline we will encounter as we move from our old system of standards to the new Common Core Standards. Without careful investigation and gradual implementation of the Common Core, students will have gaps in their understandings when the new assessments are fully implemented.
The Kansas Department of Education has recommended that teachers of Kindergarten and First Grade begin implementing the Common Core Standards in 2011-2012. The rationale is that those students will never have to take the current state assessment and this would alleviate part of the gaps we might experience when we are fully implementing and assessing the Common Core Standards.
At Essdack, we are working to make teachers’ first steps into the Common Core meaningful by providing days for investigating the content at their grade level. We also recommend investigating the crosswalks between the current state standards and the Common Core. As teachers learn where the Common Core and current standards overlap and differ, they will be able to design instruction that should lessen the gaps that might occur when the new assessments are implemented.
“Sum”thing new to start digging into!
What’s Behind the Common Core Math Standards
The release of the Common Core State Standards (CCSS) is a milestone in the standards movement that began more than 20 years ago when the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School. NCTM, along with the National Council of Supervisors of Mathematics and other mathematics organizations, supports the goal of the CCSS to describe a coherent, focused curriculum that has realistically high expectations and supports an equitable mathematics education for all students.
The Standards for Mathematical Practice, which describe expertise that math educators at all levels seek to develop in students, is also a key component of the new Common Core Standards movement. These practices rest on key “processes and proficiencies” with longstanding importance in mathematics education, including the NCTM process standards and the strands of mathematical proficiency from the National Research Council’s report Adding It Up. The five process standards which run through all grade levels are problem solving, reasoning and proof, communication, representations, and connections. The strands of proficiency specified by “Adding It Up” include: adaptive reasoning, strategic competence, conceptual understanding, proceduaral fluency, and a productive disposition.
Other key elements in a student’s success in mathematics are: • Making sense of problems and persevering in solving them. • Reasoning abstractly and quantitatively. • Modeling with mathematics. • UsIng appropriate tools strategically. • Attending to precision. • Looking for and making use of structure.
So, what is behind the common core standards for mathematics….RESEARCH!
Math: The Universal Language
Let’s face it; math is the same everywhere. What other subject can say that?
In a very simplistic example the reason for math’s universal nature is easily explained: one thing is true of all people in all nations of the world. Barring a freak nuclear reactor incident or in the case of my good friend, Bentley, who lost a finger in a battle with a pig, we all have ten fingers and ten toes. Our counting system is based upon this number. Kids the world over are taught to count by raising one finger at a time….watch for it….it doesn’t vary from country to country. The properties of the geometric shapes are the same the world over. The fundamental principles for the manipulation of numbers, the order of operations, is an accepted practice throughout the universe. The words are different, but the math is the same!
So, if all that is true why do we HATE math in America? It is NOT socially acceptable to be bad at reading ….no one ever says, “I can’t read and I’m doing just fine in life”, but it is screamed from the mountaintops that people don’t get math, never got math, can’t help their kids in math, hate math, see no reason to spend time doing math, and furthermore, don’t ever use math! No one is embarrassed to say these things aloud! There is no social stigma attached to these statements like there would be if we were referring to reading. How has this happened? We literally have a math pandemic in America and it is hurting us as a nation.
I feel really strongly about the need for teachers to educate themselves in the area of mathematics and the best practices for teaching it to our children. We, as teachers, are products of the system of instruction that we lived through. We had math “malpracticed” on us and we (including myself in the early parts of my career) have carried on the malpractice. We learned math as isolated skills without connection to other concepts or real-world application and wondered why we were doing it or when we were ever going to use it. We learned math by memorizing procedures, often without understanding, and then when there got to be too much to memorize, we quit taking the subject.
So what do we do to turn this around? It’s all up to us: the life-long learners, the people who love working with kids, the idealists who want to make the world a better place to live. We have to first recognize in ourselves the need for new information. We, as professionals, have to be willing to learn new ways to help kids understand math conceptually. We have to immerse ourselves in math content beyond the grades we teach, so we know where the math we teach leads. We have to work together with other teachers to develop a cohesive plan of instruction for all grades, based upon research into best practices, a common set of standards and a common vocabulary.
We need kids to believe math is fun! We need kids to believe math is important! We need kids to believe math is relevant! We need kids to believe math is not something to be feared! In order to make all those things true, teachers have to believe it first!!!!!!!!