I’m Seeing Dots!

Do you number sense?

What I’m trying to ask is how comfortable are you with numbers? As children, we learn to count, 1, 2, 3… Then we begin to realize a one to one correspondence, that one isn’t just a number, it is a value, a penny or one fork, or one person. The bigger the number, the more items. We learn some rules for numbers: That one plus anything makes one more. That if we take away one, it makes the number smaller. Next thing you know we are adding and taking away all kinds of numbers. Then along comes place value and we learn to carry numbers; there are ones, tens, hundreds, thousands, ten thousands, whew! And for many children, each number becomes a solid thing, no longer fluent, no longer one plus one plus one makes three. Three is, well, three. It means a certain amount, and we no longer see it as individual items making up a whole.

I would like you to try an experiment.

Without counting, take a look at image 1.

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How many dots do you see?

Now quickly look at image 2. (Remember, no counting!)

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How many did you see this time?

Now think about how you saw the dots in each image. How did you group them or add them?
Which picture was easier for you? The neatly grouped dots or the randomly spaced ones?

Show the pictures to someone else. They will probably come up with the same number you did. Ask them to describe how they saw the dots, grouped them, made sense of them.

Was their method different? Did you like it better than your method? Will you see the dots differently the next time you look?

Whether you realize it or not, you have just made some new connections about the number eight.

Now solve for x:

4x = 3 + 3 + x

Did your new sense of the number 8 make the problem less complicated? Or did you pull out some algebra skills?

We rarely see dot cards used outside of first, second, or third grade here in the US. I think we may be losing valuable math fluency by not continuing to expose our children to these practices on a regular basis throughout their education. It is especially relevant now, as teachers are asking students to think about the different elements of math problems, asking them to make connections from one set of circumstances to another, expecting them to be able to break apart and re-form numbers like a set of Lego blocks. Yet, from my own observations, we have 7-12 grade students who lack a basic ability to do this.

We give them formulas that have no meaning, theorems for which they do not understand the proofs, and polynomials they struggle to factor or multiply, because they lack the ability to construct and deconstruct numbers. Algebra is difficult because they cannot see that x can be any number. In that exercise above, could there not be more than one right answer? Shouldn’t our kids be curious about that, check for that? (Bravo if you did!!)

Cathy Humphries uses dot card exercises with her 10-12 graders. She has this to say:

“Dot Card Number Talk Commentary:
Cards with configurations of objects, that we often call “dot” card number talks, establish important new principles for mathematics classes. While it may seem that these arrangements of shapes are only for young children, we have found that they are critical for older children – even high school students – because they help to lay the groundwork for changing how students think about mathematics. Dot cards do not suggest procedures that students are “supposed” to follow; instead, they encourage students to think about what they “see” rather than what they are supposed to “do.” This frees up students for learning new ways of interacting in math class.
Some of the things they can learn from dot card number talks:
• Just as people “see” things differently, there are often many ways to approach any mathematical problem.
• Explaining one’s thinking clearly is important. This requires that students to retrace the steps of their answers and learn to use academic language, where possible, to describe what they did to solve the problem.
• It is important for students not only to explain what they did, but why their process makes sense. In the case of dot card number talks, this involves where they “saw” the numbers they used. In the case of arithmetic operations, it involves understanding the mathematics that underlies any procedure that they use.
• The teacher’s job is to ask questions that clarify what the students see rather than how they “should” see.”
Cathy Humphreys.

Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.

Introduce a new fluidity into your classroom. Check out more about dot cards in this piece by Math Coach’s Corner.

If you have good number sense activities resources, I’d be happy to link to them here!

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A Reply to Why Johnny Can’t Tell Us Why

The author writes: “Johnny (a.k.a. Mary, Bobby, Dashawn, Jaynaya, etc.) can’t think.  He doesn’t have the basic mathematical understanding of how the operations work, the nature of numbers, and the fundamental “rules” of the game of math.  She doesn’t have the “self talk” skills to decide what to do when she doesn’t know what to do.  He doesn’t have the confidence to just read the problem, take it one step at a time, and TRY. She doesn’t have any tools in her problem solving toolkit aside from learned helplessness and the response, “I don’t know” when posed a question.” http://www.rimwe.com/the-solver-blog/41.html

The link I’ve posted is the author’s full blog post on this topic. From my own experience, I believe it describes what is happening in high school classrooms across our country. The author asks the question, “What strategies or techniques have you found that are helpful in trying to turn the tide? Well, I’ve spent my summer trying to find some of the answers this author is asking. Here is my response.

You were in my class last year, weren’t you?!! Same background, different ethnicity. These students required that I fed them everything and when I didn’t, they fired me as their teacher. I have spent my summer looking for answers to your questions. I believe I have found some powerful answers. I even started a blog to talk about some of these solutions:

Inquiry, task oriented learning;
Mathematical thinking;
Visualization;
Number sense practice as part of every class;
Encouraging growth through mistakes;
Small group discussion;
Questioning that encourages students explain what they are thinking.

Oh, wait, you wanted something to deal with the anger and the apathy. That is a much tougher question.
I think students have gotten the message that they are not good enough – at math, at English, at any class. I think the response is frustration, after all, wasn’t school supposed to give them these skills? They showed up for class, they did homework or reports or worksheets. Why is it now not good enough? I’d be angry, too.

We could spend years blaming, wishing, and wringing our hands, but let’s not.

You and me and the teachers (and the parents of these children who are frustrated and unhappy with school) who are faced with this scenario are going to have to work with what we’ve been given. As for me, I am going to meet the kids where they are, use number sense puzzles and practice (as simple as I need to go, first grade level if necessary) and begin teaching these kids a new way to think about math.

Will I have to re-earn their trust? YES! It won’t be easy, but for ANY of this talk to be of any more use to our children than anything else the education community has done, for our kids to make it through these new assessments (which we really need to rethink, but that’s for another post); for our kids to make it into colleges, to have good lives, to be good citizens, we are going to have to change the message we are sending our children. They are good enough! They are clever enough to learn ANYTHING! School can be enjoyable and rich and kids can like math and literature and science and history. Anyone who tells you not to expect it, is damaging what the experience is supposed to be. Will it be struggle and hard work? Yes. And that is perhaps where we have let them down the most.

We are afraid to let our children struggle. 

Did you know that every time you have to solve a puzzle, work out a problem, or struggle to master a skill, your brain grows? Research shows that synapses fire every time this happens. Mistakes, failures, do-0vers are NOT BAD! These are things that have to happen for us to learn. For me, that means giving my math students rich, complex questions to think about, to examine, and discuss. And maybe solve.

The methods listed above are a start: Belief in our kids, funding schools, giving teachers the knowledge, as I have gained this summer, to teach kids through problem solving – not rote memory and regurgitation – and stopping the insanity of using test scores (we can look at student’s faces and observe their behavior – a focused, well-behaved student, eagerly digging into a lesson because they are interested!) to see if they’ve learned anything is what will ultimately make a difference for our children’s education.

I hope to address some of the topics I’ve touched on here in future posts. I welcome any comments, experiences, resource/research links, lesson ideas, etc. that will expand upon these conversations. Thanks for reading.