# ELL Math – 35 Weeks In

Previously: ELL Math – 3 Weeks In

In between then and now:

- We’ve talked about fraction operations to decent success, decimals to mixed success, comparing numbers & comparing fractions to great success, and translating between English and Numerals (ie: fifty two thousand five hundred and six = 52,506) to decent success
- I have a group of students that I’m trying to prepare for Algebra I next year, who’ve been exposed to: order of operations, simplifying expressions by combining like terms and with the distributive property, solving 1-step equations, and writing linear equations from a context (ie: Joe has $5 and makes $2 per hour = 2x + 5).
- Less than half of the students I have now were with me at the start of the semester: many students have left, many students have joined. My class needs to be flexible enough to adapt to the changes.
- We’re still doing Khan Academy, which exposes students to other topics I haven’t taught as a full group: mixed numbers, simplifying fractions, etc.

Basically, at this point in the year, students have been exposed to a ton of different content and lots of different strategies to solve whatever math problems come their way. One of my favorite things is to see all the different strategies in action – one student may have latched onto a physical model for multiplication while another student does a repeated multiplication strategy; some students combine positives and negatives using the number line while another group may use a physical model by drawing open and closed circles. It’s been pretty fascinating watching all of these strategies play out and seeing which students have become comfortable with the procedural short-cuts that we try and move students towards (when comparing fractions, make the denominators the same and just compare the numerators) versus the students who still need that conceptual foundation to solve the problem (when comparing fractions, draw the two fractions and compare which one has the greater shaded area).

So, here’s a snapshot of a quiz I gave last week showing where we are and the strategies students are using to get here:

Some Comments:

- I’ve started splitting the class into A, B, and C groups so I can differentiate content and problems, which is why there are 3 different versions of the quiz. This helps me with a lot of logistical things I was trying to figure out in terms of only having one gradebook but several different levels in my room.
- Somewhere in there is a student who solved the comparing fractions problems by making a common denominator and comparing the numerators, but solved the ‘least to greatest’ problems by drawing the models. I wonder if that was intentional – if that student felt the models were easier for the least to greatest problems – or if that student didn’t realize that the former strategy can also be used for the latter problems
- Teaching students to create a common denominator has made me appreciate the difference between these two questions: “What is 18 divided by 2?” versus “2 times what equals 18?”. The latter question is the one that my students struggled with more than I had initially expected but, in retrospect, I see now that this latter question requires a different type of trial-and-error thinking than the former one, but you need this type of thinking if you want to solve the least to greatest problems in a procedural way. It made me re-think how I would teach multiplication next year – I would try to include more of these types of questions earlier to reinforce the relationship between multiplication & division.
- I’ve realized that the commutative property of multiplication is not as obvious as I’ve always believed it to be, especially when you create a model to represent the multiplication. I’ve had some good one-on-one discussions around this point.
- I’ve had a shift in opinion about the role of multiplication tables. I didn’t like that using tables removed some of the cognitive burden from all future multiplication problems or that, frankly, students would just use a neighbors table instead of making their own. Multiplication problems became an exercise in looking up data in a table rather than an application of a conceptual understanding of multiplication. I used to think that these tables were a crutch that students should be steered away from – instead, if they need to multiply something, draw a picture or make a model to solve the problem.But, when I would tell students to do this, I started to see is that students weren’t realizing that they can re-use their models for similar multiplication problems. For example: I would watch students multiply 7 x 6 by drawing 6 rows of 7 dots, then move on to multiply 7 x 7 by drawing 7 rows of 7 dots without realizing they could just add one more row to their previous drawing. And, since the commutative property of multiplication isn’t immediately obvious, 3 x 7 and 7 x 3 would result in two different models. This reliance on models and the time it took to draw them was starting to discourage students and distract from the other topics we were discussing (ie: adding & subtracting fractions with unlike denominators).
So, instead what I’ve started doing is: students can use premade tables during class, but they can’t use them on quizzes or tests. However, I will give them a blank table which they can fill in as part of the time it takes for them to complete their quiz. I like this system because they are still doing cognitive work and convincing me they know how to multiply when they make the table. This is also a totally valid strategy for any other test they take in any future class – if you need a multiplication table, take time to make your own.

- I discovered that the Marcy Math Pizzazz worksheets, while generally below my rigor expectations at the high school level, are a decent quick resource for additional problems at the middle & elementary school level. They are especially helpful because they allow for self-checking. Several students prepared for this quiz by working through problem sets from that book. It also makes my differentiation efforts a lot easier to manage.

So…. there are some updates on what I’ve been up to. Lots of fun.

Do you encourage students to use the approach they are most comfortable with when they solve a problem? Or are you trying to get them all to use the same approaches by the end of the year?