Directions: Using the Integers 0 to 9, each only once, how many different ways can you fill in the blanks to make the statement true?
So, Christopher (@trianglemancsd) posed this question to twitter over the weekend:
Now, this weekend I was out in the woods hunting deer, so I got this notification while trudging through a cedar swamp. I quickly jotted down a reply that first hit me, put the phone in my pocket and trudged on, but of course that is never enough with Christopher…
DOH! Yes Christopher, it is. I’m in the WOODS, leave me alone! So when I came in for dinner I tried to construct a better response..
It’s not the most mathematical definition, so I was wondering what my students would say… This was awaiting them when they entered the room today:
I haven’t had a lot of time with the group I currently have to work on constructing good thinking responses so I didn’t know what I would get, but here is what I got.
The -0.13 Camp:
Overall, this group considered “rounding up” as only “making the digit bigger”, and had their mind blew when I asked how rounding up was making it negative 0.13? There was a long moment of pause; ideas flashing across their features as they struggled with this concept. Many became unsure of their answer.
The -0.12 Camp
Overall, this group was confused by the rounding “rules”. Many explanations would not lead to correct rounding for positive numbers and these students need a quick refresher. There was one student who understood what to do and took that into consideration when rounding, going to -0.12 because to round up was to make a bigger number.
The “Other” camp:
I am not going to post a picture of these responses, but these students had answers other than -0.12 and -0.13, and had major errors in their mathematical thinking about rounding or just guessed.
It was really had to discuss this question without imposing my idea of what the answer should be. I had many students ask me what the correct answer was throughout the discussion. I told them that was what we were trying to discover, and would not tell them my answer until they all agreed upon their way to round this number. They were confused with this concept at first because it following their rules did not produce what they expected- but only when that was implicitly pointed out to them. Many did the mechanical procedure for rounding and didn’t examine the number or it’s implications.
Once we finished our discussions, each group came to the same conclusion. That while they want math to be consistent- this did not appear to be until you considered the concept of negative. They initially wanted the procedure to be the same, by using the terminology of rounding up they wanted the number to be larger. Then they moved into the number line and comparing the distance from specific numbers. Since this was a half number- that caused a little more discussion about which way to go. They decided to round it to -0.13 because it would remain consistent with their concept of rounding, but with reflection around 0. Since a number would be rounded up in the positive, it would “round up to more negative”.
I challenge those of you who read this blog to introduce the question and discussion to your students, and blog about it. There was a lot of great mathematical thinking that happened today.
So I found this blog and this article got me thinking about the current direction of testing.
A growing number of studies conclude that students perform worse on tests when they take them online than when the questions are on paper.
A study published by MIT and conducted at the U.S. Military Academy found that the students who did not use computers scored significantly higher than those who did.
The researchers suggested that removing laptops and iPads from classes was the equivalent of improving the quality of teaching.
The study divided 726 undergraduates randomly into three groups in the 2014-15 and 2015-16 academic years. The control group’s classrooms were “technology-free,” meaning students were not allowed to use laptops or tablets at their desk. Another group was allowed to use computers and other devices, and the third group had restricted access to tablets.
“The results from our randomised experiment suggest that computer devices have a substantial negative effect on academic performance,” the researchers concluded, suggesting that the distraction of an electronic device complete with internet access outweighed their use for note-taking or research during lessons.
The research had an unusual twist: the students involved were studying at the West Point academy in the US, where cadets are ruthlessly ranked by exam results, meaning they were motivated to perform well and may have been more disciplined than typical undergraduates.
But even for the cream of the US army’s future crop, the lure of the digital world appears to have been too much, and exam performance after a full course of studying economics was lower among those in classes allowed to use devices.
“Our results indicate that students perform worse when personal computing technology is available. It is quite possible that these harmful effects could be magnified in settings outside of West Point,” the researchers concluded.
The Hechinger Report reported that writing online essays may contribute to a widening of the achievement gap.
The U.S. Department of Education launched a study of fourth graders using computers for writing compared to fourth graders using paper and pencil.
High-performing students did substantially better on the computer than with pencil and paper. But the opposite was true for average and low-performing students. They crafted better sentences using pencil and paper than they did using the computer. Low-income and black and Hispanic students tended to be in this latter category.
“(T)he use of the computer may have widened the writing achievement gap,” concluded the working paper, “Performance of fourth-grade students in the 2012 NAEP computer-based writing pilot assessment.” If so, that has big implications as test makers, with the support of the Department of Education, move forward with their goal of moving almost all students to computerized assessments, which are more efficient and cheaper to grade.
In the study, high-performing students — the top 20 percent of the test takers — produced an average of 179 words per assignment on the computer, three times the number of words that the bottom 20 percent produced. They also used spellcheck, backspace and other editing tools far more often. The researchers found that these high-performing students were more likely to have access to a computer and the Internet at home.
But these high achievers were in the minority. More than two-thirds of fourth-graders’ responses received scores in the bottom half of a 6-point scoring scale that rated grammar and writing quality. Overall, the average fourth-grader typed a total of 110 words per assignment, far less than the 159-word average on the 2010 paper test.
In looking for explanations for the disparity in performance, it seems likely that the high-performing students are more familiar with computers than low-performing students or even those in the middle.
But it is also likely, at least to me, that it is easier to read and re-read a passage when it is on paper than to read it online. Some young children may have difficulty scrolling up and down the page.
And there may be a difference in recall associated with the medium. That requires further study.
Let me confess that I have tried and failed to read books on a Kindle or similar device. It is easy to lose your place; it is hard to find it again. Maybe the difficulty is age-related; after all, I have only been using a computer for 32 years and began using it as an adult. Children who grow up in the digital age may not have the same visual problem that I have in reading large blocs of text. But it will take more studies to figure out when it is beneficial to use the computer and when it is not. Unfortunately policymakers have rushed into online instruction and online assessments on the assumption (untested) that there are no downsides. They do this, as the Hechinger Report says, because the computer makes it easier and cheaper to grade tests. Standardization has some benefits. But it also has drawbacks. We should be aware of both.
Write about a time when work felt real to you, necessary and satisfying. Paid or unpaid, professional or domestic, physical or mental.
Surprisingly this prompt could be quite lengthy and complex for me- so I will trim it down a bit and talk about the first time work (nonacademic) felt necessary and satisfying.
I grew up on a small family farm, there was plenty of work to go around. We had cattle, chickens, pigs, ducks, geese, lambs and more. We had many jobs to do for feeding our animals and upkeep of their shelters and grazing land. When you are a 8 year old boy there isn’t a lot of that type of work that feels necessary or satisfying. Dad knew this- he fought to get my brother or I out doing chores, until he became smarter than the average bear.
Our main livestock was cattle, and it takes a lot to maintain them. We had to put up hay every summer- trying to get 2 crops in each summer. We had to fence in pasture and maintain it. We had to give them shots, take care of hooves, sew lacerations, assist with births. When the calves come, you have to clean them and feed them- first with bottles and then pails. Hopefully mom will take over the feeding duties- if not you have to take care of them 3 times a day, every day. As an 8 year old, none of “activities” are fun, interesting, or satisfying. They are just necessary.
Dad put a twist on that. One fall he asked if I wanted to go with him to the cattle yard. I said sure- getting away from home was a rare occurrence. We went to this place that held a large metal shed, and corrals that seemed to stretch forever. There was so much going on I couldn’t take it all in. We found a spot in the auction hall and watched the sale of cattle. This held my interest for about 10 whole minutes until I got bored and started keeping myself busy- counting rafters, examining how the shed was built, finding patterns in cowboy hats, etc. This all was all washed away when I recognized the next bull that was led into the arena, it was ours. I was absolutely mesmerized with the bidding process on our animal, and the next 29 after that. People were paying us money for our animals, and paying well. It left quite an impression on my young mind. On the way home Dad told me “Next year you are going to own your first cow, you will take care of her and you will get to sell any of the calves she has.”
This blew my mind, I was going to get a cow and I got to sell the calves I raised for once- and get money. This opened up a whole new world to me.
Later that fall we went to another cattle sale and I got to look for my new calf. My calf. All of a sudden I became an expert on cattle, I wandered the yard looking at calves. I was judging them by appearance- how clean they were, how the acted in the corral, how big they were- I was such the eagle eye. I finally picked one and we got her at a good price, her name was GiGi.
From then on, any work around the farm was necessary and satisfying- I knew the importance of keeping GiGi fed, sheltered and kept safe. I no longer complained about chores and even took initiative to do them on my own. That was my first experience of ownership, and hopefully I can help my students find theirs as my Dad did for me.
I work at a school that services a juvenile center, as such I get to see the product of the bad side of the modern family unit every day. I have had many people ask me how I can teach here, and work with these kids. The fact is: these kids need me, they need someone who is open to listening to them- no matter what they have done before. The bad side: many of these kids don’t think they have the power to change their lives.
Case in point- there was a student here who lost hope, he figured that he would never be released from the center. We talked about things, he seemed to understand and actually talked rationally about situations and what he could do to change the outcome. The thing is- he never believed it. One thing we always talk about is how running from your problems only delay the outcome, and typically the punishment becomes harder and longer because you try to avoid it. He seemed to understand and talked a big game. I thought he was actually making a change.
He ran today, and has not been seen.
Directions: Using the integers 1 to 9, using each number only once, fill in the boxes to create a polynomial and its factor.
What is the other factor?
What patterns do you notice?
Can you expand this to a polynomial of degree 4?
Lately I can not stand sitting at PLCs or PD where fellow colleagues complain about students and their basic fact skills. So with my Master’s in Special Education- Learning Disabilities classes I looked into it a bit…
There has been a “back-to-basics” movement in mathematics ever since the release of the Trends in International Mathematics and Science Study (TIMSS) showed children from 20 countries outperformed our nation’s 8th grade students (Peak, 1995). The focus of this back-to-basics push has been basic mathematical operations; addition, subtraction, multiplication and division. Statements similar to “the student will increase the ability to complete division facts (1 to 9) from no understanding of division facts to the ability to complete division facts (1 to 9) 50% of the time through group instruction and use of manipulatives/computers” often appear in Individual Education Plans (IEPs) for students with learning disabilities (SpEd Forms, 2016). One thing that has not been addressed as prevalently is the existence of mathematical learning disabilities (MLD), its role in student’s ability to perform these math operations and what adaptations and interventions students need to be successful (Mazzocco & Myers, 2003). In an attempt to answer the question of how mathematical learning disabilities impact a middle school classroom; this literature review will examine what MLD is, some causes of MLD, how it impacts mathematical learning and how to provide interventions to implement in the mathematics classroom.
Review of the Literature
The process of learning mathematics places a high demand on a student’s cognitive processing. Mathematical abilities are cognitive skills and are frequently used to measure a student’s cognitive ability (Riccio, Sullivan & Cohen, 2010). When deficits in math are identified and represent a delay in developmental functioning or the ability to process information in one or more of the mathematical domains, that student suffers from a mathematical learning disability (also known as dyscalculia). Since mathematics encompasses a wide range of skills with varying levels of complexity, the ability to identify and define specific core deficits of MLD has not been possible (Geary, 2004). A reason for this is the fact that different domains of function have been identified with poor mathematical achievement, including reading levels, working memory, spatial visualization and executive functions (Mazzocco & Myers, 2003). Geary also states that “the goal is further complicated by the task of distinguishing poor achievement due to inadequate instruction from poor achievement due to an actual cognitive disability” (2004, p. 4). As such, a measure specifically designed to diagnose MLD is not currently available. Current practices use standardized testing combined with measures of intelligence (IQ). A single score from these tests do not imply that a student has MLD, but consistently low scores over multiple academic years often indicate some form of cognitive or memory deficit, and a diagnosis of MLD is warranted (Geary, 2004).
There are numerous ways researchers have attempted to classify dyscalculia, most referencing how math skills are taught in an educational setting. Mathematics is typically broken down into two classifications: calculations and reasoning. Math calculation is the application of algorithms, computation and fluency; math reasoning is the ability to assess a situation and determine what tools and steps are needed to solve it (Riccio et al., 2010). Students presented with mathematical problems will be expected to flow fluidly between calculation and reasoning. Dyscalculia disrupts this flow by inhibiting a student from accessing knowledge or context based on the type of mathematical learning disability. According to Korsc (1974), there are six types of dyscalculia dependent on the mathematical processes involved:
- Verbal dyscalculia: difficulty naming amounts, digits, numerals, operations, terms
- Practognostic dyscalculia: difficulty using manipulatives or pictures
- Lexical dyscalculia: difficulty reading mathematical symbols
- Graphical dyscalculia: difficulty writing or manipulating math symbols
- Ideognostical dyscalculia: difficulty in understanding mathematical ideas and mental calculations
- Operational dyscalculia: difficulty in performing mathematical calculations
Students can fall in one or more of these categories, and many students with MLD also have a second form of retrieval deficit (Geary, 2004).
Geary states that dyscalculia occurs in 5% to 8% of students in school. Ricco et al. (2010) and Geary (2004) both state that MLD is a familial disorder, with diagnosis rates 10 times more likely than families in the general public. While this may appear to be a genetic disability, there are also environmental factors that can affect mastery of mathematical skills. Students from low socioeconomic communities are more likely to suffer from mathematical learning disabilities. Since mathematical knowledge is primarily acquired through academic settings, one has to consider whether curriculum, educational opportunity or teaching practices play a role in student achievement (Riccio et al., 2010).
Neurology also plays a role in dyscalculia, with studies indicating that multiple regions of the brain are accessed when performing mathematical operations. The pariental lobe is responsible for spatial visualization, and is activated in many mathematical functions. The intraparietal sulcus performs calculations and comparisons; it also converts words into numbers. The right inferior parietal lobule, left precuneus and left superior parietal gyrus all are critical when performing subtraction calculations. Two parietal areas, the intraparietal sulcus and posterior superior sulcus are all active when students have to estimate and approximate. Research has also shown that the areas of the brain required for exact calculation are the same as those required for language processing, providing connections between reading learning disabilities and mathematical learning disabilities (Riccio et al., 2010).
Early identification and intervention for students with dyscalculia is important. According to Rico et al., most children with math disabilities in grade 5 will continue to perform in the bottom 25% of students in grade 11 (2010). Venkatesan & Vasudha affirm this by stating “there are seldom cases of middle or high school level students who could ever perform certain higher level tasks” (2014, p. 93). Geary also observed that the ability to retrieve basic facts does not improve throughout the elementary years (2004). He goes on to state that many children with MLD do not show a shift from procedure-based to memory-based problem solving. Riccio et al. (2010) also observe that students with dyscalculia have difficulty in acquiring new procedures or correcting existing ones. This implies that teachers need to identify the learning needs of children early and adapt their curriculum to meet these needs. Teachers need to allow students to use the tools they have and build new understanding and procedures using those tools.
Geary states that most cases of mathematical learning disability also involve a reading disability or attention-deficit disorder (2004). Teachers have to be aware of how these secondary disabilities interact with a student’s MLD. Learning strategies students employed for reading literacy should also be used in the mathematics classroom. Interventions would include oral accommodations, large print text, alternative note taking strategies, graphic organizers and the use of technology. Manipulatives, alternative seating or learning areas, and segmented class periods should be employed for classes for students with an attention-deficit disorder. Accommodations for homework, assessments and performance should be integrated into the daily routine of class in order to provide opportunities for success.
Mathematical learning disability, or dyscalculia, is an important learning disability that is often overlooked and mishandled in today’s classroom. The causes of mathematical learning disability include heredity, environmental and developmental factors. Although early diagnosis and intervention are paramount for student success, accurately identifying if a student has MLD and what type is difficult. Data obtained from a single year is not enough to determine whether a student has MLD, students must show a continuing deficiency over multiple years. Many students with dyscalculia also have other learning disabilities, such as reading disabilities or attention-deficit disorders. In order to adequately support these students, teachers need to build upon the student’s prior knowledge and procedures since students with MLD typically inhibit new algorithms. Students with learning disabilities consistently perform in the bottom 25% of the population throughout their student careers. This is not a topic that can remain hidden behind the lack of fact fluency, teachers need to address students’ needs so they can be successful in the mathematics classroom.
Bottom line, if you are going “back-to-basics”, then you are not helping your students. Help them in their skills by giving them appropriate work that incorporates those skills and give them the support they need to complete it. Back-to-basics is not a movement anymore people, it’s an excuse.
The paper I wrote is here, with the references included: