Math and Numeracy Learning Colleagues,
I would like to start a discussion here about challenges in teaching mathematics that you have encountered in adult basic education. My main reason to do this is that I believe each of us struggle with some area of mathematics and I know we can support and learn from each other. We haven't had this type of discussion before but I hope that those of you who are silent will find a moment to share from your valuable experiences.
Here are some questions to help us get started:
- What has been the biggest struggle for your learners this past year?
- What content/topic/lesson, do you wish you had more time/education/resources on in adult education?
- What are some of your major roadblocks to success? ("If I didn't have __________ then I know my learners could __________")
- What has been your biggest success this year?
These are general questions but I want to hear what your experiences are/were. I plan to share mine as the discussion grows but for right now - I want to hear from you.
I look forward to your comments. If you don't want to post them publicly you can always contact me via email, too.
Brooke Istas
istasb@cowley.edu
Comments
1. The biggest struggle for my learners in the past year is for the low level math students (3.0 or less) to remember the names and characteristics of different quadrilaterals.
2. I wish I had more computer resources for geometry.
3. The biggest roadblock to my success is financing. Working in a corrections setting there is little or no money for computer programs or manipulatives.
4. My biggest success this year is getting my students to think beyond the surface when it comes to math, i.e.. learning just for the test. They are beginning to ask deeper questions, some of which make me think.
My biggest challenge is that students need to learn years' worth of math ... in a week or two.
I wish I had more resources that had more conceptual instruction along with 'here's a bunch of problems to practice!' (I just did a google for the concept of divisibility. It's all about 'divisibility rules' -- even the ones that explain what it is, explain in terms of "if you put it into your calculator you'll get a decimal part of the answer," which is helpful... but not as helpful as explaining about putting things into groups...)
My biggest success this year has been getting just a little (but it helped a lot) better at ... oh, sounding nicer, and being more aware of the nonverbal stuff of communication that can trigger anxiety and 'brain shut down.' I'm pretty sure that one of my ladies improved a ton this year because I'd learned her silent messages and that what I thought of as cruising the room and seeing who needed help "felt" like I was the Inspector On Duty, Watching for Mistakes. Making a point of 'noticing' things right on those tours and then moving on made it easier to make corrections in a helpful way... and that I needed to correct and *move on, * not hover (while oh, that other person would be snatching out the calculator if I did that, but if I stuck around, and enforced things, he learned that actually, he *was* capable of figuring out things on his own).
For a positive integer N, if we can find natural numbers K and L such that K*L = N, then we say that K and L are factors of N, or we may say that K and L are divisors of N, or we might say that N is divisible by K and L. If we are given N, how do we go about looking for such complementary factors K and L? Often we choose a number H < N and then divide, maybe using a calculator, to see if the quotient N/H = Q is also a natural number. The "divisibility rules" that you find on many websites (and in number theory books) are simply "quick" methods to decide if certain small numbers H do in fact divide N exactly, without actually carrying out the division N/H. The easiest cases are to try 2, 3, 4, 5, 8, 9, 10, 11. Every time I go to the gym and find an empty locker, I check its number and factor it (to help me remember it), so I definitely use a few of these elementary divisibility rules. I am delighted when I get 207 = 9*23 or when I get 187 = 11*17.
Of course any question of finding factors of a number N should lead to considering prime numbers. A number P > 1 is prime if it has only the trivial factorization P = 1*P. Notice that if N = K*L and K <= L then K <= sqrt(N), so when looking for a factor of N you need only try the prime numbers that are no larger than the square root of N. If none of these primes is a divisor of N, then N is itself a prime. If you find a prime P that is a proper factor of N, then you can continue by factoring N/P. Student exercises: find the first 12 primes. Is 701 prime? Is 703?
You can think about finding factors of N in a different way, namely as looking for where N appears in the usual multiplication table. Lets think of making our multiplication table using a "spreadsheet": the rows are numbered from top down as R1, R2, R3, ... and the columns are numbered from left to right as C1, C2, C3, ... and in the K-column and L-row we have the entry K*L. The integer squares 1, 4, 9, 16, 25, 36, ... appear on the main diagonal in the table. Primes appear only in R1 or C1. There are lots of things to notice and play with in this table. If we add all the entries in R1 from 1 to N, that is,
S(N) = 1+2+3+...+N we get N*(N+1)/2. To see this just write the sum forwards and then in reverse and pair the numbers and add:
{1 + 2 + 3 ...+ [N-1] + N} + {N + [N-1] + ...+ 3 + 2 + 1} = (1+N) + (2+[N-1]) + (3+[N-2]) ... = N*(N+1).
If we then sum all the entries in R5 we have 5*1 + 5*2 +5*3 +... +5*N = 5*N*(N+1)/2. And if we then add all the entries in R5 out to the main diagonal entry 5*5 and then up column C5 to the top we will get 5*5*(5+1)/2 + 5*4*(4+1)/2 = 5*5*5. The same is true for the numbers in each of those L-squares with vertex on the main diagonal. We always get the sum is the cube of the row and column number.N^3. After convincing ourselves of this by computing some, we then think about adding up all the numbers in the whole upper left 5X5 square. This should be 1^3 +2^3 + 3^3 + 4^3 + 5^3. But it is also (1+2+3+4+5)*(1+2+3+4+5) = (5*6/2) * (5*6/2) --- a square! So we have found a formula for the sum of consecutive cubes. Wow. Will your students be amazed and impressed at what they have found?
Yup! I do use the multiplication table a lot for the visual discovery of the squares. Our teachers also include at the beginning of the student notebook with a 25x25 grid of times tables, which comes in really handy with ALEKS for the students who have essentially no math background (our "text" is a binder with notes and pages and problems with space for them to write in, which is better for teaching tho' students sometimes begrudge that they can't re-sell it at the end of semester).
One of my big challenges for those students is to get them to think about what the numbers mean, though; often they're simply copying the squiggles. That's why I wish I could draw well enough to make a Harry Potter-esque "cloak of divisibility" that woudl shake out a quantity of objects (which students could choose from, just for fun) into groups of the divisor named to see if things shook out evenly. I think it could also be used for seeing the equivalence of equivalent fractions, which is another huge issue for abstracting. The student I was working with yesterday still, still struggles with the idea of 2/5 being 'the same' as 8/20 ... after all, shouldn't (in her mind) 8/20 be *lots* bigger???
Modumath explains it starting with quarters and dimes (but, alas, she's taking the placement test too soon and doesn't think she has time for actually watching the videos :( )