The Standards: Is it really thinner?
Going from California to the CCSS-M standards does have “less”. They’ve taken out a lot from integers. Students don’t have to do any operations, they just work on concepts and value like comparisons, etc. They still have to do the same amount of work in fractions (addition, subtraction, multiplication and division). The problem is that the lack of operations in integers really hinders having them do things like resolving simple algebra equations. For example, to solve 4x + 4 = 12, you need to subtract four from both sides of the equation, something more easily understood when you know how to add/subtract integers. It also means that you can’t really do problems with integers like 4x + (-4) = 12. Since integers are pretty integral to pre-algebra, and early algebra it hampers that development. Others have decried the “rolling-back” of algebra for all, and I don’t know if I want to join that bandwagon since I never thought much about exposing all eighth graders to algebra. Now to the next part. Having kids doing all operations in fractions is problematic, and will continue to be an issue. I hear folks say no worries, they will be getting more fractions earlier so they’ll be ready by the time they get up to you. Yeah, that was how the early algebra curriculum was supposed to work, and I’m not impressed with how that turned out (lots of kids taking algebra two and three times to pass).
Next, what does this mean?
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Assessment tasks: Going for complexity, getting complications
I like having my kids write about and defend their problem solving. They’re eleven years old, this is a good age to start that process. I won’t speak to whether this is developmentally appropriate or even possible with younger students, or whether this will work with ELs just learning the language, for my kids it’s do-able.
There is a real love for multi-step word problems, and at a certain point, it’s about hoop jumping and not higher-order thinking. Complicated is getting mistaken for complexity. My take is that a lot of the “experts” doing these tasks is they get taken away with “rigor” (another word I hate) and make things a lot more complicated than they have to be.
Here is an example from an assessment task was about deriving ratio from a task on making bead bracelets. There are two types of bead to use, glass and spacers. There are three types of glass beads (A, B, C) and three of spacers (D, E, F).
Design a bracelet using at least two types of glass beads and one type of spacer bead.• Use between 8 and 12 glass beads.• Use at least 6 spacer beads.• Use no more than 25 total beads in your bracelet.
Instructional Shifts: Modeling vs. Algorithm
If I never read another FB status update with some parent who is a math/science major/professional de-crying that their child is not being taught the most intuitive or easiest way to solve math problems…well, I can dream can’t I? This argument pre-dates Common Core. Anyone remember the Washington weather woman doing a video taking down “Everyday Math”? I have a father who was a Mathematics major, and I barely slogged my way through the Algebras. I figured out math in college and after taking statistics courses (and more cognitive development in the analytical part of brain). My dad had little patience for “new math”, but no worries, my teachers were all old school and it was a struggle for me. We cannot base our mathematics curriculum on whether parents will be able to help their kids with homework. This comparison of private vs. public education shows what happens when your curriculum is based largely on parent desires, instead of best-practices. I think parents should have a voice in the general goals of education (what are we preparing your child for, what values do we teach them, etc.) but teaching methods should be waded into carefully by non-experts.
I have no problems with using some of the new models coming out like tape diagrams, etc. but they have limits when you’re dealing with larger and more complex numbers such as we find in sixth grade curriculum. Here is an example. We’re given a lot of examples of using tape diagrams to model and teach division of fractions. That works really great for simple problems, but they have to divide mixed numbers (whole numbers with fractions). Tape diagrams, not so good for that. Sure enough, I peaked at one of the new “aligned” texts and it showed tape diagrams for fractions, then the next lesson had mixed number problems using the standard algorithm. It was especially bad because it just threw the algorithm in with little prep or background, with no understanding that this is a complicated multiple-step operation, and the tie between it and the tape diagram method, unless explained, will likely not be inferred. That’s more a curriculum complaint, but it’s a problem.
Also, there is a love affair with number lines. Number lines are great, but I’m remembering the former love affair in fraction teaching for pie charts, and the lovely description of this by an Chinese educator as, “You Americans use too much round food to teach math.”