Several of you may already know that the College and Career Readiness Standards for Adult Education have been released!!! Listed in these standards on page 48, are the Mathematics Standards, "Standards for Mathematical Practice". There are 8 of these practices:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitivatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning.
I would like to have a discussion on this topic of mathematical practice because they are essential in making sure that math is more than rote memorization or using some formula. It's about developing our learners into more than just shadows of our instructional practices but about creating mathematical thinkers. Therefore, let's each pick one of these practices go to page 48 read about it then post in your own words what that mathematical practice means in your own words. Make sure it is clear to other adult education practitioners and perhaps discuss how you teach (or would teach) this practice in your classroom.
I hope you will participate even if you have never posted before; we have so much to learn about these new standards and mathematical practices. Let's do it together!
LINCS Subject Matter Expert
Math and Numeracy
Should be, should be, should be... is this reflected in how the students are assessed?
Lots of my guys break down at number one: making sense of the problems. If the class / situation is typically structured, then they don't have to really make sense of it; they just apply the procedure they've been taught in that section to whatever numbers are in the problem.
My health care math folks seem to have gotten pretty good at figuring out which numbers matter in a situation, though, and picking from their personal library of strategies to figure out what to do. They've learned an "I have... I need" process that's usually setting up a proportion, and they know to set up the proportion with things that match. I really like the approach and just might have to chase down the teacher and find out more (I'm a tutor and help the students working on the problems).
I also find that the students in my ABE math class (many of whom are GED or HSD students who come in for math remediation) have a great deal of trouble with the first practice, "make sense of problems and perservere in solving them."
We're finding that the "making sense" part often has root causes in not understanding the vocabulary, especially terms that are used with specialized meanings within mathematics and word problems. Again, if they are using a traditional text book, they are used to getting by without actually understanding the terminology--they are just following the example at the top of the pg. So I don't use text books. And we are starting, as a departmental professional learning community (PLC) to work on the "Vocabulary Acquisition and Use" standard across subjects, including math: "variable"| is a great example word in this respect--it has multiple distinct though related meanings: synonymous with "changeable" in standard English (and economics), but referring to a unknown constant in algebra. It's important to draw students' attention to these types of words and phrases.
The "perseverance" aspect has to do with morale and a sense of efficacy. What I see is that my students need to experience the frustrations of not knowing what to do, followed by the joy of figuring something out for themselves. This needs to be modeled, encouraged with group work, and reinforced by having students find their own "aha" moments in solving new problems, or in recognizing where their reasoning went wrong on a problem they've already worked incorrectly. Many students don't realize that even (especially) "good" math learners go through periods of not knowing how to solve a problem, but there are strategies that may help, though none are guaranteed "recipes."
I've begun to really emphasize careful reading/paraphrasing, estimation skills, guess and check, etc. as strategies.
The more carefully we observe where students go wrong or give up, the more fascinating it gets! And the better we will be able to help students help themselves.
- Wendy Hoben
Berkeley Adult School
I am *totally* experiencing this this morning. We're in a three week summer session and there's a nutritition course happening, and they've got an assignment analyzing fast food meals and their nutritional content. They've got the pre-algebra prerequisite done... but do they understand that to answer the question "does that meal provide one third of the recommended fiber" means to divide the recommended fiber by 3? Well, yes, if I point it out ... and figuring out the percent of the kcal that are coming from fat?
They *are* getting it... but I'm leaving my drawing of the part-and-whole fat/whole meal idea of "percent of teh whole thing" on the board. Maybe by the end of the class, with more practical examples, they'll start to grasp the relationship between ".38 of" and that it's a part of the whole thing... happily, the both of them recognized that 2100 was going to be a whole lot more than 1/3 of 2300...
In observing several training sessions involving imlementation the math CCSS in our state, I noticed several of our instructors having trouble with the significance of the Stds for Practice (as did I) ; so, I began searching for simple explanations of exactly how the 8 Standards applied, and ran across these short presentations about each of the 8 Standards on the following McGraw Hill website:
Perhaps those of you that peruse this discussion may have no issues with grasping the finer points of the standards for practice, but I can assure you that many of the folks teaching math in Adult Learning Centers lack a total understanding of the significance of these 8 standards, and could benefit from the presentations on the afore mentioned website.
Area 3 Regional Associate
KY Adult Ed
Thanks, Brooke, for highlighting these significant practices. I have just a couple thoughts I'd like to share.
These Mathematical Practices grew out of the work from NCTM's Process Standards work (math as communication, problem-solving, connections, reasoning, and representation) and the National Research Council's Adding It Up work with the five mathematical proficiency strands (procedural fluency, adaptive reasoning, conceptual understanding, strategic competence, and productive disposition). The point of each of those documents was to encourage teachers to do more than just teach isolated procedures and rote computation.
I had hopes that the mathematical proficiency strands - especailly clearly stating that we need to do more than just teach procedural fluency - would catch on in the classroom. If it is happening, it's doing so very slowly.
My fear is that these new practices, while built on sound research and principles, will again take a back seat to memorization of rules, especially since these 8 Mathematical Practices are not written as clearly as the NCTM and NRC documents are.
However, I am forever the optimist and hope that we as teachers realize that our learners find it challenging to see the difference in these problems:
4/5 = 8/10
4/5 + 2/3
4/5 x 2/3
4/5 ÷ 2/3
Unless we help students develop into mathematically proficient adults who can apply those Mathematical Practices, students are going to continue to struggle understanding why the rule is to sometimes cross multiply, sometimes turn one of the two fractions upside down, sometimes perform the operation across the tops and bottom - and other times to not do so. With an understanding of the different meanings of fractions and operations, students can develop a set of strategies that they can test out, adapt if one strategy isn't working, persevere to work through the problem, then reflect on whether the answer makes sense or not.
As an optimist, I am hopeful that teachers will embrace these Mathematical Practices.
Noting that one of the mathematical practices calls for us to communicate math ideas clearly and with precision paying careful attention to math lingo, I think we should make sure that we do that when writing about math.
In that spirit, I have a few quibbles with the note by Donna Curry. None of the 4 lines of math expressions she wrote is a "problem".
"4/5 = 8/10" is a claim that something is true, a proposition. If there is a problem here it is "Why do you think, or how can you prove, that these two fraction expressions evaluate to the same result?".
The next three expressions or formulas, are not statements, claims, propositions, or problems. They are simply high-level recipes for calculating something. Each one calls for the use of a different basic operation: addition, multiplication, division.
4/5 + 2/3
4/5 x 2/3
4/5 ÷ 2/3
Here I suppose we will want to evaluate the expression, that is, actually carry out the described calculation. Of ocourse we first need to know whether we wish to carry out exact arithmetic with a fraction as answer, or are we going to do decimal arithmetic with a specified number of decimal places in the answer. (Mentally, or using pencil and paper, or calculator, or computer?)
Students know that they use very different processes to add, multiply, or divide integers, and the same is true for decimal numbers. So there is no surprise that to add, multiply, or divide fractions requires three very different processes as well.
Through my assistantship, I have grown very fond of the standards and math practices for K-12. Even though some might appear difficult to tackle they really are not. The one that might be tricky is the Reason abstractly since there is not as much time to ensure students have a deeper understanding as well adult basic education rarely touches proofs. My favorite is "Model with Mathematics" since is requires educators to guide their students on how a particular mathematics concept is applicable in the real world.
Now for some educational sites you will see how the standards align with a particular lesson. I hope that the same insight will be applied to math related courses for adult learners.
I agree that it can be difficult to assess understanding of reasoning within the time constraints of a class and without having students work on mathematical proofs. One of the tools that I've found immensely helpful is questioning their reasoning, such as "Why did you do that?", although I have to say that I've learned to tell students that I often ask why even if their answer is correct. Otherwise I found that many of my adult students would assume they were wrong and start changing their answer as soon as I asked "Why?" Sometimes I'll ask why they used particular numbers in a problem - what was the reason for using those particular numbers? It can also help me identify misconceptions that students have and help them modify their thinking more quickly rather than just showing the correct process to get to the right answer.
New students to my class are often initially very uncomfortable with this, because they are used to just trying the four basic operations with any two numbers in the problem until they get something that matches a multiple choice answer. Over time though, they become more comfortable, and actually experience pride when they feel confident that they can explain a problem all the way through to me. I find it really rewarding to watch that change, as well as knowing that I am reaching into those standards of reasoning and communicating mathematically.
Thanks for the link to the McGraw Hill videos on each of the 8 practices. I've watched two so far and feel they are quite useful. I'm going to strongly encourage the volunteers who help in my ABE math class to watch some of them, especially the first.
Here are some interesting sites/sources I’ve been using to look at the Common Core MPs:
http://map.mathshell.org/materials/tasks.php - includes lesson plans that follow a very interesting model (initial assessment, response, revision, group work, metacognitive analysis, etc.) and which include samples of (middle and high school) student work
http://insidemathematics.org/index.php/common-core-standards - includes classroom (K-12) videos highlighting the mathematical practices, along with tasks that can be used to explicitly address specific MPs
http://www.illustrativemathematics.org/standards/practice - videos of students and tasks that can be used to explicitly address specific MPs
Math Professional Development Coordinator
CUNY Adult Literacy and HSE Program
101 W. 31st Street, 14th Floor
New York, NY 10001
Thank you for sharing some of the resources above for unpacking and understanding the standards. I wanted to include a description of the standards provided by a 4th grade teacher on the Scholastic website. http://www.scholastic.com/teachers/top-teaching/2013/03/guide-8-mathematical-practice-standards I have found this beneficial just to quickly reference for a better basic understanding of each standard.
I am struggling with fully understanding the nuances of #7 and #8. I realize that they are very related standards and both deal with recognizing patterns and structure and generalizing. It seems that 7 is more based on students using the patterns they see to increase efficiency. So a student multiplying 9*2*7 should multiply 9*7 first, because doubling at the end is easy. Or a student who in order to do the problem 49*8 does 50*8 first.
Standard 8 focuses on finding those patterns to develop general methods and shortcuts. There is no need to make a general method/shortcut for 9*2*7, that seems to be a problem of personal preference. However, noticing that when you multiply by 10 then you add a 0 to the end is a shortcut or recognizing special products like (x+5)^2 or (x+5)(x-5).
I do not know if that makes sense... probably not, as it does not yet fully make sense to me!