Both my grandmother and my mother are excellent cooks. My grandmother is a classic 1950s chef--she has a huge repertoire that includes foods like meatloaf, chocolate cake, and apple sauce--all from scratch. My mother is a classic New York Italian cook. Just thinking of her stuffed shells or artichokes makes me salivate--I'm literally doing it now as I type.
The most interesting (and key) difference between my grandmother and my mother is not their cuisine, but rater the way that they cook. While my grandmother lives and dies by her recipes, my mother cooks by feel and taste. My grandmother has all her recipes written out and she follows them step by step, ever faithful, never deviating. My mother doesn't have anything written down, and if you ask her how to cook her artichokes, it's not guaranteed that you'll end up with the same results, because she tweaks and adjusts as she goes. This is why replicating the tender leaves that melt like butter in your mouth is only possible if you've cooked them right alongside her, to learn by doing with her.
Both my grandmother and my mother are good cooks, and can perform. But the key difference in the way that they cook is that my mother is much more adaptable. She can change course mid-meal to bring the sauce back to the way she wants it to be. My grandmother, on the other hand, must rely on a specific set of instructions, and if things go awry, she either needs to scrap it all together and start over, or live with a sub-par outcome.
I share this story of two cooks because it has real relevance to math education. Many of us were taught that math is a set of rules and steps. If we just follow the steps and rules, we're guaranteed to get the Right Answer™. However, as I discussed yesterday, there has been a movement in math education to give kids an opportunity to engage with mathematics, not just memorize its rules and properties. The way that we have been taught is much like my grandmother's cooking: follow the recipe and don't deviate. The way that math is being taught now is akin to my mother's cooking--learn the way that cooking works, and you'll understand why you should lower your burner if your meatballs are burning on the outside and still raw on the inside.
From the outside, this way of teaching math often seems needlessly foolish and complicated. It seems to take longer and be more confusing. But often those criticisms come from folks who have already decided in their minds that, as a matter of principle, math is confusing. I can tell you from being in a classroom, that if you don't have that already decided, engaging with math can be a fun exploration ... an adventure!
Standard for Mathematical Practice #1 gets to the heart of this issue. The standard states that students who are proficient with mathematics will make sense of problems and persevere in solving them. The full description is below.
This standard describes a student who is proficient with mathematics--it is a goal for students to grow into as they develop and learn. So if your 3rd grader isn't exhibiting ALL these traits, take a deep breath, let the stress melt from your shoulders, and then let the breath out. It's ok.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Let's think a little about the task I posted on Friday. (For a refresher, check out Sunday's debrief.) It was meant to address a 3rd grade content standard. And within that task, I can see a number of elements from the above paragraph:
- looking for an entry point (a student would need to figure out how to divide 18 by 3, in both contexts)
- analyze givens (the two contexts), analyze relationships (the questions in the task that ask them about similarities and differences)
- younger students might rely on concrete objects (the multi-link cubes!) to help conceptualize and solve a problem
Did you notice that with that task, we weren't asking a student to just memorize division facts? Rather, we were asking the student to play with division, see it in different contexts, make observations, and learn by doing. I don't know about you, but this is definitely a shift from the way I learned division. But I see, regularly in the classroom, how powerful that shift can be.
Suggestion: Take a look at some of the math work your child is doing. Do you see evidence of SMP #1? It doesn't need to be embedded in every single thing they do ... but it does need to be present. Feel free to share back ways that you see SMP #1 playing out in your kiddo's math learning.
Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October! This week's Tuesday night math chat will be on Twitter, at 9p ET--follow along using #CCSSM!
