I teach math. I don’t want to teach it anymore. Instead, I
want my students to learn to think about math. The distinction is important. And Beautiful.
I have found myself standing at the intersection of two ideas. Each one describes very clearly the most important activity of my classroom:
…letting students struggle with and form math knowledge.
The first idea is from Intentional Talk, E. Kazemi & A. Hintz
I want my students to engage in conversations about math: feelings, ideas, suppositions, beliefs, stuff they already know, stuff they only partially know, stuff they think they know, but don’t. And then I want these conversations to lead them to want to know more. I want them to argue and convince, to persuade and be persuaded, to either hold their stances, or change their minds.
In short, I want to facilitate growth: knowledge, courage, conviction. I am not alone. Research has shown that mathematical conversations are a powerful learning tool for our students. I want to teach my students to wield that power, to feel the joy in discovery; the confidence in being sure of what they know and can accomplish.
The second idea (and its intersection with the first idea), is Inquiry Based Learning (IBL). IBL is not new, and the style, in fact, appears to trace back to Socrates. A fellow named R.L. Moore popularized it’s use at the college level, and it is popping up all over under all sorts of names and in all sorts of complicated guises. The goal is the same, however: to get students thinking, talking, and making mental connections to facilitate real understanding of the material.
“Understanding is critical to long term retention” paraphrased from Understanding by Design, by G. Wiggins & J. McTighe
Obvious statement, that. Yet, many of the students I have encountered don’t remember matter they’ve been “taught” for 24 hours, much less for the end of course test. Re-teaching, spiraling, test review – you won’t meet a teacher, myself included, who fails to build these into the lesson. That leads me to think that we are not successfully teaching for understanding, but for students to memorize and regurgitate: wash, rinse, repeat.
What about the children who cannot memorize?
The diagnoses: “Short term memory is lacking… doesn’t have the language skills to form long term memory… computational skills are weak….” The list goes on – and those are just for the students who have been identified! I have come to the conclusion that memorizing isn’t the way to teach anything!
Don’t get me wrong. I am not saying children shouldn’t remember. Memorizing isn’t remembering! True remembering is recalling something you have learned, putting it to use, twisting and turning it to solve a puzzle. It should not be cumbersome and painful. It should be a source of satisfaction. Memorizing – now that can be painful!
I love to learn new things, new words, new ideas. I love to learn.
My students don’t. By the time a student walks into my math class – 9th, 10th, 11th grade – they have already “fired” math as a place where they can learn anything.
“I hate math… I am no good at math… I’m gonna fail this course…”
Okay, I’ve got them talking! Now to turn the conversation 180 degrees (sorry- couldn’t help that one!)
Start with intentional talk. Give your students three sentences about math to discuss. Group them anyway you like. Explain the rules. Practice the first one with them (model the language, let them practice!), turn them loose and walk and listen! Ask clarifying questions, if you must, but otherwise stay in the shadows. (I’ve noticed that when they can’t see me, they forget I’m in the room. Lol)
Make this a regular habit for discussing the Big Ideas of math. You will learn all kinds of things about how your students think!
Next, incorporate IBL. Instead of a set of statements, present your students with a “situation.” (This can be anything from a simple linear expression to a table of numbers in a sequence.) The key is to keep it visually simple and straightforward.
The following example can be used to help students consider the relationships of numbers, and learning to use variables appropriately in explaining relationships, as well as an extension into sequences of numbers.
Start with a chart like the one below.
The instructions to the students: The chart has been created according to a specific set of rules. Study the arrangement and find as many relationships as possible among the numbers.
Give your students a worksheet of the chart, and for the next 45 minutes, have the students present their findings without any specific directions from you. Students present by writing on the board or on larger versions of the chart posted around the room, drawing diagrams of their connections, or describing those connections.
While they are doing this, walk around, observe student markings, or comments. Ask clarifying questions as students present (or ask other students to re-state or paraphrase what has been said), and keep the activity moving by having students go to the board. Once there are no further ideas, summarize the student’s connections, The most obvious noticing may be the perfect squares down the middle, or the fact that there is a regular slope to the numbers on the diagonal. (The slope will allow you to ask leading questions to take them to the idea of sequences.) The key to this activity is to have the students explain the connections – to find and answer the question as to why the patterns exist.
At this point in the exercise, as students try to come up with an explanation of what they are seeing, the intentional talk strategy of listening to each other, restating, offering agreement or disagreement, and refining or changing ideas will take place.
A final write up in a journal or on the worksheet, will allow students time to reflect on the activities and learnings of the day. For more information on creating these lessons, see The Open-Ended Approach (NCTM).
K.I.S.S. – B.N.E. !
Keep it simple, straightforward – but NOT easy. There should be multiple steps involved in the process. Keep the instructions simple, too. In the example given, the instruction is to “notice all the possible patterns” and “compare/share with your partner.”
Using the same techniques of facilitating intentional talks, walk and listen, question to clarify, and ONE MORE THING: mentally pick out those ideas you want to have shared with the whole class. Ask students if they would be willing to share, and pick on them in the order you want to introduce the ideas. Have big sheets of paper on walls or tables, encourage lists and illustrations. Encourage students to rephrase what other students are sharing.
Don’t take over the conversation!
Be very careful with your comments. The minute you become the “expert” and not the question-er, your students will look to you for the “right” answer. They become afraid of exploring their ideas, of re-arranging words and knowledge. Remember that this conversation is crucial to the sense-making that is going on.
You may find that you need to clarify things if they become stumped, but only to help them get started again. Try prompting students to rephrase or ask questions of each other to open up more ideas. Here are some “question stems” posters like the one below
to post on your wall. Don’t be afraid to work through a student’s idea, even when you may not know where it’s going! It’s about modeling curiosity and learning behaviors and how to go about finding answers!
DO work through every situation you want them to explore; anticipate their responses and misunderstandings!
DON’T be afraid to redirect the conversation, or travel down side paths – but keep your end goal for learning in mind. You can always revisit these conversations later!
This is messy stuff! Your students are going to be noisy, happy, excited (about learning and having a voice), and maybe a little nervous. Math class isn’t supposed to be, well, fun.
You are going to have to get out of the way. Facilitating learning isn’t as easy as it looks. There will be times when you will want to give them the answer so badly…. But just don’t.
This takes time. You may find that things take a little longer to get through, but you will also find that you don’t need those endless days of review. AND the spiraling takes care of itself because your kids are actually using the stuff they’ve already learned to craft new connections.
Pssst! Did I mention this is what student directed learning looks like? And wait’ll you tell them about 20Time and GeniusHour!!!!!!
As always, your comments, lesson shares, and feedback are deeply appreciated. Here we come, 2015-2016 School Year!!!
For more on what this looks like in the classroom, check out Steven Strogatz’ post