As I have mentioned a couple of times, I am on a committee with both high school and higher education faculty that is working on lessening the number of students who require remediation when they enter college. We were talking about how students end up in the "remedial" mathematics classes and one of the college faculty made the remark that if a student couldn't multiply a 2-digit number by another 2-digit number by hand that he or she would automatically end up in the lowest class. In my mind, that seemed rather harsh, especially since calculator use is highly encouraged in our classrooms and have been since middle school. Why in the world would a 4th grade skill automatically preclude a student from progressing to a higher level math course when technology is so readily available and expected to be used? So, I asked. The college faculty member replied that it has to do with transferring from numeric to algebraic. If a student can multiply a 2-digit number by another 2-digit number, he or she can use the same process to multiply a binomial by a binomial - it is the same process. It has to do with number sense.

Later in the meeting, a different college faculty member made some of the same points. (He didn't come in until later in the day, so he had missed the earlier comments.) He mentioned fractions, signed numbers, and decimals as the basic skills that students should have and that students needed to understand the concepts behind them. Like with multiplying, these are concepts that resurface in mathematics. I have certainly seen this as we have worked with rational expressions. If students don't understand how to work with numeric fractions, how are they going to transfer that to algebraic fractions?

So now I am sitting here this evening starting to rethink calculators. On the one hand, I have students who have struggled with basic facts and are (mostly) able to work with algebraic concepts using the calculator to handle the computation. But am I doing my students a disservice by providing them the calculator? What if I didn't allow them calculators at all at the beginning of the year and tried to remedy the issues that came up? Would I even get anywhere? Maybe what makes more sense is as I am introducing the concepts that tie into those earlier concepts - introducing multiplying polynomials, for example, that I begin with the number sense concept that preceded it. It would take some research on my part - although with the Common Core State Standards, the progressions have been fairly well-documented. By taking the time to (briefly) review the numeric concept, maybe students would have a better handle on the algebraic concepts. It's a thought. I need to think a little more on this one.

## 5 comments:

Last time I had to teach division of polynomials I spent a lesson or two going over long division. I think the time was very well spent and enabled many students to make that leap. Would those same students have managed to master division of polynomials without the prep? I don't know. Would I do it the same way next time? Absolutely.

I recently saw a post (which I can no longer find, but I did look for it), where someone was talking about teaching kids to add fractions in a more algebraic sense. So when asking a question like 3/5 + 2/3 instead of getting the answer 19/15, they would try to give the answer in terms of the numbers 2, 3, and 5 only. So the answer would be (3*3 + 2*5)/(5*3) which helps them to build the algebraic skills necessary while still working with familiar numbers. This should probably be done when they first learn to do common denominators and such, but there's no reason not to do it later as well.

This is my second year of teaching. Last year, I encouraged students to use calculators for anything and everything they needed them for. Someone else pointed out to me that lack of numeric literacy may contribute to difficulties in Algebra so I resolved to make that more of a priority this year.

This year, I spent the first couple of weeks of school on basic number skills - operations with integers, fractions, percents etc. To be honest, I'm not sure it helped - they still use their calculators for simple operations. Also, when we started talking about operations on rational expressions and I asked questions about how to perform operations on numerical fractions I was met with mostly blank stares. Maybe if done in a different way it would work better

@Alex - it is David Cox's post I believe you're talking about: http://coxmath.blogspot.com/2012/04/comma.html. He posted it right after I had done rational expressions. I am definitely looking at that next year.

@Leah - Thanks for sharing what you did last year. I'm still thinking about what I want to do about it. The more I think about it, the more I am leaning towards integrating the numeric reviews as I introduce the algebraic concepts, similar to what Anonymous mentioned above with long division and division of polynomials.

--Lisa

My experience has been that if students get to higher algebra and still don't have fractions down, they've built up a block. Students who have mis-learned, misunderstood, or just been frustrated with the same topic over and over again just can't be retaught that idea in the same way they've seen it before. Every year teachers have tried to reteach it to these students and every year when the students can't get it, they feel like failures and lose all confidence in their ability to understand these numerical processes. When I try to directly reteach fractions I have no success. But these students have advanced to algebra 2 or beyond. They can think about math abstractly. So I've had a lot more success reteaching fractions or numerical operations through algebra. We discuss canceling and why it works in pre-alg or algebra 1 and then go back and look at how reducing of fractions worked. I've had a lot of success with this because then we're not trying to reteach the students something they're sure they can't learn and don't understand. We're teaching them something new which they're interested in learning, and they can then go back and realize they actually do understand that thing they thought they could never get, and better yet, it wasn't a frustrating process. When I teach this way, I haven't had too much trouble with calculators because it's not 5x6 I'm asking of them, it's how reducing works and how we can apply it to variables, then to numbers which has nothing to do with calculators. On a side note, I made a polynomial division worksheet that builds it up based on regular long division. I don't know if it's any good, but it's an example of what the first comment was referring to. here's the link: http://dontpanictheansweris42.blogspot.com/2012/02/polynomial-division.html

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